Eurail pass or individual tickets

Your itinerary is working against you and is not efficient at all. Given your very ambitious (many would say, too ambitious) plan, you need to make it as efficient as possible. Some suggestions...

1. Don't fly into Paris, then go to London, then double back to the continent - that wastes a trip across the Channel, wasting time and money needlessly. Fly into London, then train onward from there - never double-back.
2. Don't fly into one city and then return to the same city after traveling around Europe. Instead, fly ("open jaws") into London, then out of your last city (presumably Rome).
3. I know, I know - you're going to say "but the flight is cheaper into and out of Paris" or maybe you're going to say "I already booked my flight and its round trip to/from Paris and can't be changed." I'd respectfully disagree. My point remains: even if you have to pay something to change your flight, or you end up with a more expensive flight, your current plan will cost you more - you are going to waste money crossing the English Channel needlessly, and you are going to fly from Rome to Paris needlessly (you lose a day, you have to check in/out of a hotel, this burns lots of otherwise usable time), and worst of all, you are going to waste at least 1-2 days of an already-too-short trip trying to "save" money. That's not smart.

IMHO you need to fix these more serious problems with your itinerary first (I'd also consider eliminating a stop or two, but that's a separate question). Worry about rail pass details after that.

Hope that helps.

I'l add this about your plans: before you do anything else, you need to be honest - brutally honest - with yourself, about how much actual, usable time you have. You say your trip is "21 days". Does that include the day you arrive and/or the day you leave? If so, then you actually have just 19 days (maybe just 18 days, depending on how well you recover from jet lag - I'm assuming you will be flying to Europe from North America).

If this is your first trip to Europe (just a guess...) you need to take this seriously:
No matter what time your flight from North America arrives in Europe, you can not count your arrival day as a real, usable day. Most people are too exhausted, jet-lagged and stressed out to make much of that day (try to get out and walk as much as you can in the sunshine and fresh air, to try and stay awake until after local dinner time, then hit the bed and rest). Many people are still only at half-strength the following day (helps if you are young and in good shape) so I like to factor that in, but lets set that aside for the moment.
The same is true for your departure day: no matter what time your flight leaves, all of that day will be consumed by the tasks required to get you on your flight.
So, bottom line: start by being honest about how many full, usable days you will have in Europe, not counting arrival or departure date.

OK, so lets say you have 18 to 21 days there. You have (I think) 7 cities on your itinerary. That's at best 3 nights in each place, which means at best 2 full days in each. That's physically possible, but most people would suggest it's not a good way to see these cities, and not realistic.

Remember: you will not instantly teleport yourself from one hotel room to another in a distant city. The basic logistical tasks required to move from one city to another take time. More time than you think. With your itinerary, you end up having very little time in each place.

With 21 days (or 18 days, as I suspect you may really have), I would cut some stops from your plan - I'd do 4, maybe 5 cities, tops (and that's still moving fast). Consider this:

1. Fly into London
2. Train to Paris
3. Fly to Florence
4. Train to Rome
5. Fly home directly from Rome

Save Amsterdam, Munich and Lucerne for another trip.

Hope that helps - good luck!

I agree with David's comment, re-arrange your locations in as near a straight line as possible, and fly open jaw into the first and out of the last.

As for the pass, do not buy a pass at least until you have read this webpage: https://www.seat61.com/Railpass-and-Eurail-pass-guide.htm
Especially the section "Eurail or point-to-point tickets?". It is very likely that it will be a lot cheaper just to buy normal tickets for each of your rail trips, most of which you can get big discounts on for advance purchase.

• Your only trip in the UK (assuming you start in London) is the train from London to Paris. This is run by Eurostar (website: https://www.eurostar.com/rw-en/? ). They are currently selling tickets for mid-May starting at €50.50
• Paris to Amsterdam is Thalys (website: https://www.thalys.com/ ). Mid-May is currently €65.
• Amsterdam to Munich is your longest hop, I would find a cheap air fare for that.
• Italian trains are cheap (website: http://www.trenitalia.com/tcom-en ). Florence to Rome is priced starting at €19.90

21 days (20 nights) ÷ 7 locations = 3 nights per location. That is doable, but a rush. Each change of location you will lose ½ day. London, Paris and Rome need more than 3 nights. I would drop one or two locations, probably Amsterdam or Munich as they are outliers. High Speed train will get you from Paris to Luzern in 4½ hours.

Good advice. Thank you! I'm rethinking... May have a question or two later, but this helps.

Consider this for planning purposes. I left my rented apartment in Paris heading by train to Nice. The train departed at 9 or so in the morning, with an anticipated 4+ hour trip time. Someone walked into the train (killed him big time) and we were delayed until 9 pm that evening. Stuff happens. Simplify, simplify, simplify....and bring extra money.

If you get a pass, no need to get a 4 country pass. Keep in mind that not every one gets jet lag, unless you are sure you will be jet lagged upon landing. With 21 days I look at the night train options, say from Amsterdam. From Amsterdam take the day train to Düsseldorf, then the night train Düsseldorf to Munich. In your place, that would certainly be my option. If you cannot do that, then think about dropping Munich.

Munich dropped! You and I think alike! Still working on other pieces, but this is helpful. Thank you to all.
Don

Don't forget flights. I spent 5 weeks (35 days. day 1 arrival at 8am, day 35 departure mid-day) working my way in a loop from Paris to Rome to Amsterdam and back to Paris. We stayed in 10 locales, from 1 night in Carcassonne to 5 nights in Rome and Amsterdam, and every number in between (4 Provence, 3 Florence, 3 Cinque Terre, and so on...). What made this different than your proposition is that we rented a car for 2 weeks, so travel time was reduced, each destination was 2 hours travel from another (except 3 by train from AMS to PAR), etc., and, our destinations were not all major cities, as yours are (so, arrival in Provence meant parking the car, not getting through a train station to public transport to walking to our apartment). That said, arrival in the Cinque Terre and Florence by train was easy. Both are small (Florence, relatively). Rome, Paris and Amsterdam took much longer.
I did not feel rushed, and in fact, felt like the timing was almost perfect. My husband, who had no part in the planning, was very happy with the timing. The way I sorted that out was to sketch in my mind every.day, in every.place. So day 1 Paris; what are we doing. Day 2? I typically plan a for-sure outing in the morning (a site I absolutely want to see, and a second for the afternoon that is more flexible - so I have something, or a few things, in mind but don't necessarily get tickets or reservations in advance. So mornings were Eiffel, Colisseum, Rijks, Vatican, Abbey de Senaque, hiking in the CT, etc, and afternoons were the Duomo climb, Roussillon, biking Vondelpark, swimming at Pont du Gard, and so on. (we were traveling with 3 kids ages 9-14, so there is that affecting my planning too).
As you imagine these days unfolding, you will have a better sense of how much time you have in each and whether you will be able to experience all that you dream of experiencing. Then you prioritize places, as you are already doing. So, yes, travel can take ½ day, but if times correctly, this can serve as a rest time, and time twitch the scenery of the country out of the train window, rather than be seen as a 'lost half day'. I appreciate the travel because I get to sit for a half day and just 'see' the country. Then Im ready to hit the ground running when I get to the next place.
Back to my first sentence about flights. We flew from Rome to Amsterdam for $50/ticket, and there are cheaper tickets but we didn't do it early enough. They can be $30. And I do think there is something to be experienced on those travel days besides rest; it is its own kind of immersion in place to be figuring out train schedules and studying maps, and navigating stations, and getting around. After doing this a few times with my 13 yo on her first visit to Europe, she was quickly able to do it all on her own; 'no mom, its this platform [duh]'

Thank you to everyone. Travel plans reduced, looking forward to some train travel (I love train rides), and two flights to reduce travel time and costs. ...and no, I won't be purchasing a Eurail pass.

What I didn't mention is that 40 years ago a friend of mine and I traveled Europe for 3 months with backpacks and stayed in youth hostels. I couldn't count the times we spent sleeping on a night train to save money. We would spend from 7 in the morning to 10 at night in a city or small town, catch a train, repeat. Yes, those days were different than the present.

True, 40 to 45 years ago I did the same, saying hostels, and took a few night trains. Yes, those were the days. I still do, each trip I factor in 2 night rides, except this one in May/early June with have 4 night train rides to squeeze out extra time. There are numerous night routes managed by Night Jet covering Italy, Germany, Austria, Croatia, Hungary, Prague, Poland too by not by Night Jet. They are certainly an option if one wants it.