If it were me I would just choose just 2 or max 3 cities in the same country - even in the same region of one country, and concentrate on them. Two nights is a minimum for me, in one place - any less and you just don't get to see much of anywhere - especially when you're on a budget. Trying to see three countries in 7 days would mean an awful lot of time (proportionally) spent on the routines of finding your hotel/guesthouse, checking in and out of rooms, getting to and fro train stations and airports, and actually being on transport.

Personally, I would choose Belgium, and spend 2 nights in Brussels, 2 nights in Ghent, 2 nights in Bruges, and do a day trip to one of the fabulous Belgian Monastic breweries in the countryside. You'll get history, architecture, castles, great food, great beer, great wine, beautiful scenery between the cities. English is really widely spoken, which makes things much easier if you don't speak any European languages. Really nice walking tours to be done in each city, or bike tours if you fancy that instead... the lambic brewery in Brussels is really interesting, there's a great little comic book museum (I'd only heard of Tin tin, before I went, and it was still interesting!) and when it all ends and you come home, you can feel like you've seen and done a reasonable amount in that country, and feel like you actually 'know' Belgium a bit.