Hungary is known for its Turkish baths. The Ottoman Turks conquered the area at one point in Hungary's history. The baths are very authentic and fun. Szechenyi baths are the most well-known.
Szechenyi baths are great, but they are a huge tourist attraction. If you like to live more like a local, the Margit Island baths habitate almost entirely locals, at least on the weekend. The people selling food there speak only Hungarian, because they never see any tourists. Margit Island is the big island on the danube between Buda and Pest. These baths are only outdoors, (they don't have any fancy indoor baths), but they do have a water slide and a wave pool! It's very fun, and I recommend you visit; they have the same hot, warm, and cold baths as does Szechenyi.
Hungarian is very different from almost any other language, so it would be an interesting learning experience (as well as practical when visiting the Margit baths) to learn at least a few phrases like greetings and how to order food and drink.
The national liquor there is called palinka. My favorite is cherry but they have various fruit palinkas there such as pear and peach.
You'll probably end up doing this anyway, but walking by the Danube at night (or even day), and seeing the many gorgeous bridges lit up is an unforgettable setting. The Opera House, and actually most architecture there, is beautiful as well.
I recommend the ruin pubs. They are friendly and relaxed with an interesting decor. I lived by Instant (pronounced IN-shtant), which is a cool one. A big, popular one is Szimpla (pronounced SEEM-pluh) nearby. Szimpla has one seating area with a bathtub chair.
I really love Corwin Teto (the rooftop club at Corwin mall in the summer). It's big and open-air and friendly, and they have theme nights like Reggae Wednesday.
I had a lot of fun trying out goulash almost every day. Some places have excellent gyulas, but many have mediocre. If you have just one good experience of the spicy, meaty dish, then the whole trip is worthwhile.
Hungarians also love their pastries, cottage cheese -based deserts, paprika (mmm, paprika crackers!), and sweet wines. Try them out at the local supermarket.