Spending a few days in each country is optimal
There are 28 countries in the EU, and you need to allow a day to travel between each. So, if you spend only three days in each country and allow a travel day between each, that will take 112 days.
Given that, as an American, you can only spend 90 days in the Schengen Zone you will have a hard time juggling your time. And that's apart from having a very rushed trip and seeing very little of each country.
Trains are an excellent way to get around for short trips and international trips between adjacent countries. Budget flights are best for longer trips.
This site explains train travel in much more detail that there is room for in an answer here. http://www.seat61.com/ . It will tell you how to determine whether a pass is worth the money for your trip.Sometimes a Eurail pass is cheaper than individual tickets and sometimes not; in your case, it probably is not. You already know that with a pass, you still need to make reservations and pay an additional fee on most high speed, international and night trains. Here in Italy, the reservation for a high speed train would cost 10 euro. The site also has links to the various national train sites where you can price individual tickets - it can be cheaper to buy individual tickets, especially in places like Spain and Italy. Read the section called "Common railpass mistakes: Italy..." and the section called "Common railpass mistakes: Italy..." on the seat61 site, for example. You're more likely to save money with a pass if you want to travel spontaneously and can't buy individual tickets in advance to take advantage of discounts. Do read the sections on Quotas as well. On popular routes like between Paris and Amsterdam, or between Paris and Italy or Spain, there can be a limit to the number of railpasses that can be used on a given train. If you don't make the reservation early enough, you can end up having to pay full fare even when you have a pass or you will have to find alternative routes which may require multiple train changes.
Use http://www.skyscanner.com/ to find budget flights and compare with trains.
A couple of days per country will result in your spending more time on trains/planes in transit rather than actually seeing things in the various countries.
Look at a map, pick a few cities, get hotel rooms in each city for a few days/nights, and you can visit surrounding towns of most european cities by bus or train. You can google different rails and trains to take depending on the country/city in question
By train if you can. But we're talking about what, 26 countries?
DON'T-- Europe is for the real FOLK.
If you have the time, by train.
If not then use budget airlines.
