I agree with a lot of the points made in other replies:
1) A "cheap" flight is only a bargain if it goes to, or near to, someplace you want to be. If you're going for a month or two, you can take more time getting to your desired destination; if you have a short trip, it's a false economy to waste that time to fly to a place far from what you want to see.
2) When considering flights, look at all the costs, in both money and hassle, at both the US and the European end. For instance, if a flight from JFK is cheaper than one from Albany, but you have to spend a night in a hotel in each direction to make sure you don't miss the plane, plus parking at JFK prices instead of Albany ones, most like there go your savings - not to mention the extra hassle. Similarly, If you are using separate tickets to save money (say, Norwegian to Dublin, then Ryanair to Warsaw), are you prepared to spend a night in Dublin to make sure you don't miss your flight back to the US, or to buy a last minute ticket if you do miss your flight?
Here's a good summary of the issues involved in separate tickets: https://www.tripadvisor.com/Travel-g41707-c207311/Newton:Massachusetts:Connecting.On.Separate.Tickets.html
3) While it is certainly true that some countries are cheaper than others, remember that you can find ways to spend money or save money anywhere. Supermarkets are cheaper than restaurants; hostels are cheaper than hotels; local buses are cheaper than taxis.
4) Similarly, it's not the cost of a country you need to worry about, but the specific places in that country that you are visiting. For instance, I found that prices in Liverpool, Manchester, and Glasgow were far lower than in London. Paris is more expensive than the rest of France. Etc.
If you're interested in my wonderful and inexpensive trip to Glasgow, Manchester, and Liverpool, by the way, here's my trip report; you'll see that many top attractions there are free, and unlike London, the ones that aren't free are almost all cheap: https://community.ricksteves.com/travel-forum/trip-reports/uk-trip-report-glasgow-manchester-liverpool-in-september-2016
5) There is simply no rhyme or reason to airfares anymore. I'll repeat my favorite example. In March 2017, I was looking at airfares from New York for May 2017. This is supposed to be "too late" to get any decent prices. Well, NYC to Zurich was $349, nonstop, with a choice of airlines (not a typo, three hundred and forty-nine dollars!). As a friend pointed out, taxes alone on many tickets are that much. Then I looked at NYC to Geneva. Nonstops were about $1200; the cheapest ticket was about $650 with a 12 hour layover (Royal Air Maroc was having a sale); the cheapest one-stop with decent waiting time was about $700. So, why was Zurich so cheap? Why was Geneva so much more expensive? Your guess is as good as mine.
And this story gets back to your initial point. While the flight to Zurich may have been cheap, a night in a Zurich hotel would eat up any savings, and while Zurich is very well connected to much of central Europe (there's a train station right in the airport), the train ticket won't be cheap either, unless I took a chance on a non-refundable ticket (a risk if the plane is late and I miss the train).