Firefly, if you plan to just go to those cities, then I really wouldn't bother with the Interrail ticket. Train fares are low in CEE.
In relation to Bp, you're coming at the best time of the year, but try to come on a weekend night. Check out these outdoor bars: Hold Udvar (Margit Sziget), Tuztate (Tuzolto utca), Zold Pardon (beside Petofi Hid), Szimpla Kert (Kazinczy utca 14). They all open till late and you can find the street locations on
Avocado (Nyary Pal utca 9) and Koleves (corner of Dob utca and Kazinczy utca) are great for food. Raday utca and Liszt Ferenc Ter are worth visiting for drinks and food.
Also, don't miss the baths, the best of which is Szechenyi Furdo (City Park - yellow metro line).
What is your accommodation budget? Some very cheap places available - from €10 per night upwards.