I hope the question is still relevant...
-Yes, you have to pay for the spas. There are more and less expensive ones. Széchenyi fürdő is the most popular one, also not extremely expensive.
-What do you like? I could recommend you the Opera and museums, but maybe you enjoy discos and pubs more.
Let's see, something that everybody likes... You can visit the tallest mountain in Budapest (János-hegy), there is a tower, with a tremendous view. You can go up there with a ski-lift.
-To eat: If you want to taste real local food, I would advise you to visit a market. The biggest one is located in Fővám tér, but the building is also very popular tourist attraction. I also like the market at the subway-station Újbuda-központ which has two levels, and on the second level you find places to eat.
If you are looking for something more luxurious... Borkonyha has a Michelin-star.