What is the best order of visiting these cities, and do you think 2 nights is enough? We do have the otpion of adding a third night to Vienna and Prague. Train info?

I think you are spreading yourself too thin. These are each worthy of 2 weeks. I would spend all that time in just one of these cities. My choice would be Vienna.

I've not been to Budapest, but in a perfect world I'd go there first, then Vienna and end in Prague.

If I did it again, I'd take one day from Vienna and spend it in Prague. Two nights is a bit tight in both places, but if I had the option of a bit of extra time in either, I'd spend the entire balance in Prague.

Eurail.

We spent our honeymoon in Prague and Vienna. Have not been to Budapest. My choice would be more time in Prague, less time in Vienna. There are a lot of things to do and see in both places, but Prague to me felt intimate and romantic. Vienna was beautiful, but in a big, grandiose way. I don't think you can really go wrong with any of those cities, but Prague was the most special to us.

After reading your post again, I realized I didn't directly address your questions. Personally, 2 nights would not be enough for me in any city. We spent 5 nights in Prage and 3-4 in Vienna on our honeymoon, and it didn't feel long enough. We did Prague, then Vienna, but I think I would have preferred the other way round.