Cheapest Route in Europe

First off, you may be ask to produce a return ticket at you initial immigration check. And without it they might not let you in. You buy what is called a open jaw ticket or multi-city ticket. It is a round trip ticket - just not in and out of the same city. And sometimes those tickets can actually be cheaper. I think you need to take a map, put some pins in it to get a feel of the geography. Second, read Rick Steves', Europe Through the Back Door. That should start to put some structure to your thinking. Then return with more defined questions.

You can purchase a ticket called an "open jaw" ticket that allows you to fly in to one city and out of a different city. This is cheaper than two one ways, but more expensive than a round trip from one city. As for which to fly into, you could enter different starting points and ending points into a flight search engine like google flights or Skyscanner (just make sure you check the "multi city" box when entering departure and arrivals cities, whichever search engine you choose) to check price. You can also purchase round trip domestic european flights, say fromNaples to Athens, on European airlines (this would be a separate search).

As for where to go, the list you dream of is long and will create a trip that is skimming places and traveling a lot from place to place. Better to reduce your geographic spread and maximize visits in a smaller area.
Once you start making a list of things you really want to do, some places will rise to the surface and others will lose priority. Also, train tickets to all those places is another big cost, so consider that. Once you reduce your wish list, a sensible route will make itself evident.

I agree with Frank. A multicity ticket is what you need. And a map and several guidebooks to help organize your plans. Since you're planning on a lot of train travel, familiarize yourself with this site:

https://www.seat61.com

Open jaw or multi city is what you should be looking for in flights. A good place to start would be google flights. I just did a search of NYC to Madrid with a return from Athens to NYC leaving Sep 3 and returning Oct 11 and there are several option between $650-$800 R/T

... but more expensive than a round trip from one city....... Based on our experience that is simply not true. Most of our trips to Europe are open jaw but we always price them both ways. It is rare for an open jaw ticket to be more expensive than a round trip ticket and if it is, it is in the range of $100. And many times we have found the OJ ticket is actually cheaper than a RT ticket. So you just have to price it both ways.

Kevin has a good suggestion.

You can get a cheap, less than $100 ticket, from Naples or Rome to Athens, if you plan ahead.

What time of year do you plan to travel?

As has been suggested, the territory you propose to cover really cries out for a lot more than 35 days. You mention many major cities where some of us would choose to spend a week or more, then you have travel time as you hop from place to place (more than a half-day lost with each change, on average). Finally, the more rural places you're especially interested in tend to take longer to reach than the big cities because of slow or limited public transportation. Having one day at a place like the Verdon Gorge or Chamonix is likely to cost you close to three days; you'll spend large chunks of two of them just traveling in and traveling out. I spent 89 days in eastern France last year and ultimately decided to skip the Verdon Gorge because of transportation issues--and I was already traveling around southern France. Not saying you should skip it, but that you be realistic about how much time it is going to cost you and research France to identify other great stops in that part of the country.

I would skip Athens on this trip because of the awkwardness of getting there and because Greece is so much more than just Athens. It's far better to go there when you can spend at least 2 weeks and see a bit of the countryside and one or two islands in addition to Athens.

The Blue Grotto, from most reports, is not worth the hassle of getting there--and that's for folks who are already visiting Capri/Naples/Amalfi Coast, which you don't mention as top priorities.

In my view, those two omissions are not enough to make for a maximally enjoyable 35-day trip. I suggest really digging into Italy and France with guidebooks. Figure out what you want to see in those two countries and how much time you'll have left for other destinations. You'll have more time for sightseeing in general and for outdoor adventures in particular if you rein in the geographic extent of your itinerary.

Here are some things to consider as you do your planning.

Budget - Some countries are very expensive (like Switerland), others are pretty cheap (Hungary). Major cities like Paris and Amsterdam are more expensive. Staying in dorm rooms in hostels is usually the cheapest accommodation (and the more beds in the room, the cheaper the bed) but there are drawbacks - some have limited hours of operation from mid-morning to mid-afternoon, many are not centrally located which adds commute time and public transportation costs. The more you move around the more you'll need to spend on transportation. Sometimes a rail pass can save you money, often it doesn't. For long distances, you have the option of flying. If you book tickets well in advance they are pretty cheap, but the closer you get to the date, the more expensive the tickets are. For instance, Rome to Athens on Ryanair is €128 for Sunday's flight, but only €25 for a flight in October.

Destinations - As I said, the more you have the more you'll spend on transportation. Moving also uses up a lot of time. While a 2-hour train ride may not seem like much, allow about an hour to pack up, check out, get to the station, find and board your train, then another hour to get from the station to your hotel/hostel, check in and drop your stuff before you start sightseeing. . . . which is why I mentioned that hostels may be closed for several hours during the day so you may not be able to drop your stuff. Flying takes longer because train stations are in city centers and airports are far from them, plus you need the extra lead time for airport procedures - check-in, security, boarding, disembarking, possibly waiting for luggage collection. City destinations are easier to get to because there are lots of train and bus routes. Places like Chamonix are relatively remote and can easily take a full day just to get to. Also consider that you may use your time better by limiting your countries but seeing more of the ones you choose. 10 days in Spain or a week in Hungary can give you lots of varied experiences and sights without long journeys and without the added time and stress of geting oriented to new countries every few days.

Planning resources - As others have said, guide books are really helpful. Europe Through the Back Door is a very good beginner's guide to touring in Europe. Then you'll want books for each country, maybe for each city, you are interested in visiting. This doesn't have to be a major investment. Look for them in your public library - even ones that are several years old will be a good start. When you get to fine-tuning, you may find the best information for specific sights, hostels, etc. on the internet. rome2rio.com is a good way to find routes (bus/train/plane) from point A to point B. bahn.com is the German train website, has schedules for nearly all European trains. hostelz.com and hostelworld.com are good for finding hostels at your destinations, including reviews. Some hostels have a party-all-night vibe, others enforce quiet time and attract serious travelers.

My son spent five weeks in Europe last summer. He had two different one way tickets. Both Norwegian Air and Tap Portugal sell inexpensive one way tickets. However, I would suggest that you try to figure out your itinerary first. He had intended to explore Norway but ended up buying a airline ticket from Vienna just to make his flight. He did half of his planning after he arrived. But he had one way tickets in May in the range of $200-300 one way. This was from Miami (Tap) and Ft. Lauderdale (Norwegian Air).

Flying into one city and out of another through one airline is another good option, as others have said.

Beth

It looks like you already have great information. I will just add that in terms of air travel, I often go into one city and out of another, and I've priced it and it hasn't cost more. Airline prices vary by airport location and the airline/airlines that dominate the airport. I live in Minnesota and we can get to Reykjavik, London, Amsterdam, Brussels and Paris pretty inexpensively and typically direct. In general, other places take longer and cost more. The airport that you use will have different cities (and probably more variety) that are priced lower. I often just search by cities in Europe to see what my options might be. Also, you might be able to develop a loop. Like once I flew into Paris, went to Alsace, then Burgundy, then Loire and back to Paris so there wasn't any backtracking.
In terms of Switzerland, it is all pretty expensive, for me, especially Geneva. I didn't go to Zermatt, but I heard that is really expensive, too. Luzern and Berner Oberland were my absolute favorites and I found prices a bit lower there.
Late this fall, my husband and I are going to Spain. I'm encountering pretty low prices for accommodations. Probably both the fact that its off season and its a less expensive country to travel to. Even in Madrid, I'm seeing 3 star hotels for under 100Euro and around 60 Euro in Andalusia. Because, I've been to Barcelona, my sense is that within Spain, that is a pricier place to stay.
Lastly, unfortunately, the dollar is very weak against the Euro. When I started to travel to Europe it was pretty even to the Euro, now its been about 1.2

It's all relative. We (desafortunadamente) made a couple of trips when it was at 1.5.

It's all relative. We (desafortunadamente) made a couple of trips when it was at 1.5.

I agree about flying open jaw into one airport and out of another. You are fortunate to have Norwegian Air Shuttle flying from JFK to Gatwick Airport where you can fly just about anywhere I Europe for $100 on EasyJet .com. We flew last week on Norwegian from Dublin into Stewart for $158 one way.
Remember that your first and last days of your trip are essentially wasted days. And every time you move from city to city is another wasted days. 35 days is not as long as it sounds and your itinerary may be a little ambitious. I prefer traveling in a straight line to.cities relatively close together and to cities with good train service.
I suggest skipping Athens on this trip. The following cities are easily traveled between by train: Budapest, Bratislava, Vienna, Vienna, Prague, Dresden and Berlin. These are all great tourist cities that are very lively. From Berlin, EasyJet can fly to just about any other European city you have the energy and time to see.

rome2rio website lets you plot your travel on a map. It also gives suggested pricing and routes for plane, bus, train or drive. When there is a lot of stops, I like this to see just how much backtracking is happening. There is a warning, however, and that is that the pricing and schedules are not updated for the season or pricing. It is a good general first level planning tool, and then check prices and exact times from the source's website. It will let you decide whether a cheap 12 hour bus ride is worth it over an 'cheap' airline flip of an hour.
Maybe it was due to my poor Canadian dollar, but I found trains to be more expensive than anticipated, unless purchased well ahead. Dynamic pricing occurs on all forms of transport (including more comfort levels of buses), so it will being spontaneous is not going to save you money.
I agree that this is a lot of travel in 35 days. Though Greece is somewhat out of the way, the other alternative is to put Eastern Europe on hold if Greece is more important....but actually, both Greece and Eastern Europe could be dropped and it will still be a busy trip.
If you don't know, there is a great scenic train from Switzerland into Italy - the Bernina Express. Not the cheapest method, but could fit into your plans if you drop Greece.

We found for our upcoming September trip that a Norwegian Air flight from Newburgh, NY (Stewart Airport) to Dublin, Ireland, return from Dublin a month later (we'll be going September 1 to October 3). That round-trip was $244 per person with bulkhead seats picked in advance - that particular flight is also one of the few from the US that you depart around noon and land in Ireland same day - we take off around 12:30 PM and land in Dublin around 11:30 PM (though it'll be 6:30 PM in NY), so it'll make sense to go to bed when we land (though it'll feel more like a nap!) and we're not going overnight with little to no sleep! We then have an early flight into Frankfurt on September 2nd, and return via Munich on October 2nd, then head to NY on October 3....and the Germany tickets added another $172 per person. In total we spent $416 round-trip per person. I chose Germany as our landing point since the zero-deductible full car insurance is cheapest in Germany with a Mercedes E-class automatic transmission rental.

To answer your questions, flying into and out of different cities is fairly easy to do - within Europe you'll likely buy yourself two separate one-way tickets to do that, which also allows you to go on different airlines based on price rather than airline. So far as potentially doing your round-trip from the US with one-ways - with only the budget airlines (usually only the budget airlines) offering cheap one-way fares, you'll be better off price-wise doing a multi-city ticket if you want to open-jaw your round-trip from the US itself.

So far as what you have on your list of cities - Barcelona is, by far, the most expensive on your list - I'd suggest looking at Valencia as an alternate to stay as it's lovely too and you can hit Barcelona as a day trip, or on your way to another destination as a stop mid-day to explore.

Looking at your list, you're all over Europe - your best bet is to sit with a map, or open up Google maps, and plan a route - if you'll be traveling on trains/buses, you can choose that option on Google maps and it'll note how long those options take (we rent a car, so I drive between cities) - consider that as part of your planning.....if you're traveling 35-days, you'll want to minimize your travel time, or go on the sleeper trains or buses that are available (and are pretty comfortable, and you kill two birds with one stone - getting where you're going and sleeping while you do it).

You might do well to look up some websites specific to backpacking. With a quick search I found a few that appear to provide some great information:

https://www.thebrokebackpacker.com
https://thesavvybackpacker.com
https://www.nomadicmatt.com/travel-blogs/backpacking-europe-guide

The OP wrote in another thread that his trip was August-September, so he's probably in Europe already!