I can provide you a suggestion for a hostel in Budapest if you are looking for one. I also stayed at a good one in Prague.
I have a couple recommendations for Budapest:
-Ruins Bars/the Bar on the boat (#1 & 2 according to
this list by Lonely Planet)
-
Szechenyi baths, or other thermal baths in the city. They sometimes have parties in the evening, sadly I missed one the night I left for Bratislava
-Caving (didn't have time to do that)
-Find a walking tour, they are usually only paid by tips. They have several different tours (like one on communism)
-Just walk around and look at the architecture
Also, know where you are staying relative to the Oktagon in case you get lost...everybody knows where it is lol. The tram on the "korut" (ring road) that goes to the Oktogon runs all night.
Also, I would recommend Berlin, Prague and Munich en route to meeting your friend in Milan. Even though I was sick for much of my time in Berlin I found it to be a super cool (and huge!) city. Prague was also very nice and easy to get around. Munich I found didn't have as much character as the other cities I visited, but if you are interested in cars you can go to BMW world and the BMW museum. Also, the beer is good!