A lot depends on your interests, which you don't mention. Music, art, architecture, etc. With that said, Rome is a wonderful city and full of life outside the obvious physical attractions. Northern Italy is almost a different country to Southern Italy...Florence is replete with magisterial artworks but it's a fairly dry, conservative city. Ditto Venice (except it's wet). I would add Naples to my Italian list (only two hours or so south of Rome by train). Often described as the northernmost city in Africa, it's a true experience. And a day on Capri (by boat from Naples) is unmissable.

Austria is extremely conservative. Salzburg is a chocolate box but somewhat lacking in soul. Vienna is a must if you're musically inclined for all the obvious reasons. If you're inclined towards Beethoven, Mahler etc. it's a pilgrimage and that's enough to justify a visit. Prague is full of YAPs (Young Americans in Prague), another chocolate box, lots of soul but somewhat spoiled by the tourist influx (or should I say commercialised?)

Budapest is beautiful; try to cover both Buda and Pest. The central synagogue is unmissable and the opera house has played a central role in a lot of careers (not least Solti, who started out there.) Athens, as someone else said, is a s*** hole albeit a s*** hole with the Parthenon in the middle of it so perhaps worth a visit for that alone.

Besides Naples (an obvious exclusion) you seem to be focusing on cities but the Dalmatian coast (Croatia) is extremely beautiful, inexpensive and not out of your way. The Swiss alps (not inexpensive) are also on your route and while the cities in Switzerland are entirely missable, if your journey takes you by way of somewhere like Interlaken, it's worth detouring into the mountains for a day (Wengen, Grindelwald, sometwhere like that.)

The other place vaguely in your orbit that you don't mention is Krakow, if you want to venture into Poland but pretty near the Czech border and easy to get to from Prague. Extremely beautiful city and, of course, departure point for Auschwitz which is extremely painful but probably a necessary part of any western cultural education.

Feel free to PM me if you want more info.

Edited by sydtheeagle (21 Aug 2015 8.27am)

The best place he's going to is Budapest

I did interrailing a couple of years ago and it's great. The best thing about it is you can do a lot of your travelling on sleepers (extra charges but not horrendous) which not only saves you money on hotels but is also great fun.

What I would say is you're trying to cover a lot of ground in 17 days! You don't want to spend your whole time travelling without actually seeing places!

If you're up for nightlife etc. get yourself to eastern Europe, Prague and Budapest are both cracking nights out and genuinely great cities to walk around as well. Budapest especially is fantastic. I did 30 days around Europe and it was my favourite city along with Berlin.

Edited by We are goin up! (21 Aug 2015 9.26am)

To be pedantic Prague and Budapest are considered central European.