City break? 10 days?
You definitely want to do a multi city break or you will be bored stupid by the end of 10 days.
The further east you go, the better your budget lasts.
I would highly recommend Budapest as your main destination, if thats the way you want to go. Trains are cheap - if you want to go adventurous consider Belgrade as a stop (over night train) and from there if you can stand it Saravejo is an AMAZING city.
If you have more in your budget/ less adventurous, head from Bupapest to Bratislava and then to Vienna.
Other options (considering cheap flight destinations here) Somewhere in Poland (Wroclaw is lovely and has cheap flights) then onto Berlin.
Last year we flew into Bordeaux spent a day there and then went down to the South West coast of France to the Basque area and loved it - beaches, mountains, (including sports that go with both mountains and beaches) and THE food. Its less well known to Brits than Spanish Basque. We REALLY wanted to go over the border to Bilbao and San Sebastian but just didn't have the time (we were on a VERY short break). Trains again great.
Italy is a good option, but if you want to do Florence and art galleries be ready to queue LOTS. Pisa is good for no more than a day.
Island hoping in Croatia??? Make your way down the coast, Pula, Zadar, Split, Dubrovnik?
Helsinki, ferry to Talinn, then on to Riga?