I lived in budapest for a year and loved it. there are several beer gardens in the city which are a lot of fun, the one that is coming to mind is called Romkert. There is also a great club under Budapest University of Economic Science. Most places are open till the wee hours of the morning. Make sure you check out the great indoor produce market next to that school too. The baths are an experience as well, the best one I think is Szecheny.
outside of Budapest, Pecs is nice. You can go there for a wine tasting.
Yes, the first thing to see is Budapest and the second is Pecs! The city is not too big but beautiful I think. You can take a train from Pecs to Villany (the best red wine in Hungary is produced there). You have to take a train around 2 - 3 pm because the last leaves Villany at 9 pm.
A good advice in Budapest: if you travel by a taxi, never ask the taxidriver to bring you ina nightclub, bar..etc. They get money from the bars + restaurant to bring you there and then you have to pay 500-1000 Euros for a bottle of wine. And if you say no for the waiter 2 muscoleus security men appear...
So it is better to ask people in he street!
I think the Szecheny baths in the city park are also the best ones to go to any time of the year. Also the opera house is magnificent, but you have to buy a ticket to get inside to look around. The hike up to the top of the buda side is a good trip and you'll find one of the best views around once there. I toured Parliament when I was there. It isn't very expensive and a wonderful building (tour was maybe 30 mins long). Have some falafel. Try some of the live music bars like Old Man's. I stayed at the August hostel near the city park. It was a great place to stay and run by a guy from Chicago who was very interesting. Good luck
If you want to have a big party you should go to the Hajógyári-sziget. This is an island on the Danube and there are a lot of open air party place in the summer. Every friday and saturday night there are good parties. Most of them are free, but drinks are expensive.