Budapest for 2 days, which thermal bath????

I have only been to the Szechenyi Bath, so I am no expert. Szechenyi is easily accessible and you will probably be very near it during your sight seeing so that might help make the decision. Since you are under age for drinking in Hungary, then you might want to spend the evenings in the Bath House....... :-)

Here is some information that might help you decide: http://www.bathsbudapest.com/szechenyi-bath

EDIT
Szechenyi is open 365, inside and outside. We spent a sub-freezing Christmas evening in the outdoor pools a few years back. It was a blast. Szechenyi can be crowded, or at last seem crowded, when getting in, but the place is so large that once in, it has never seemed so bad. You can buy tickets in advance which should speed up entry.
As for grungy. YES, by a lot of standards that might not be too far off the mark. We tend to leave a pretty sterile life in the US. Unhealthy? I don't think so. But if you are the least bit uncomfortable being butt to butt in a locker room then get a Cabana to change in (large private closet).

Szechenyi seemed like too much of a mob scene when we were there, so we did not go in. It was a holiday weekend though so I am sure there are quieter times. But it's the one that Rick recommends most highly. We have been to the Gellert twice, once each on two different visits to Budapest. The first time was o.k., we went in the evening and it was crazy crowded and grungy. The second time, the outside was open (this was in early May) and we went earlier in the day. I was much calmer and cleaner, a truly relaxing experience. I think the key is to time your visit and go earlier in the day, and visit at a time of year when the outdoors will be open, which presumably it would be in July. Gellert has the reputation for being more oriented toward nonlocals, but the crowd seemed plenty local to me. I thought it really exemplified the faded grandeur of Budapest itself.

I went to both Szechnyi and Gellert in late July last year. Personally, I preferred Szechnyi.

At Szechnyi, I liked the ability to move between outdoor pools of different temperatures; the weather was sunny and warm and it was a pleasure to be outdoors. I enjoyed watching the crowd around the in-the-pool chess game, and the building surrounding the pools was just stunning in evening light. I was there on a weekday evening, and although there were plenty of people there, it wasn't mobbed. Some of the pools had water jets that would go on and off, and that was a nice way to "massage" my tired body after a long day walking around Budapest. There's a very small circular "lazy river" in the center of one pool that the kids in the pool really enjoyed (and I admit, I tried it out too). I stayed until it was almost dark, and it felt like those of us who were tourists were joining the locals at their swimming pool. I got a cabana; the electric lock system makes it easy to get in and out of the cabana (and the Rick Steves book explains it well).

I went to Gellert late on a weekday afternoon, after a hot, dusty day spent up on Castle Hill, and it felt wonderful to cool off in the pool. It was fairly crowded; I would say about the same as Szechnyi. The crowd was a little more sedate, and seemed to be entirely made up of tourists. I got a cabana again, but for some reason I had a lot of trouble finding my way around the locker room. Gellert has an outdoor wave pool that some people enjoyed, but I found it underwhelming compared to other wave pools I've visited. It also had a couple of warmer indoor pools with water jets, etc.

One of the previous replies referred to the relaxing experience at Gellert. Neither place was relaxed when I was there (probably at a different time of day at Gellert), but I wasn't looking for relaxation--I just wanted a novel cultural experience, so crowds and noise weren't an issue for me.

I say, go! It's an unusual experience and, although it's not cheap, the uniqueness of the experience made it worth the cost to me. The Rick Steves book explains the procedure very well, so read it carefully before you go. I liked having a cabana for the additional privacy. Incidentally, both places will rent you a towel, so you don't need to bring one.

One thing to be aware of: some of the pools at each place require a bathing cap, even for men. At Gellert, I saw a couple of guys ordered out of the pool by an official for not having bathing caps, while their bald friend got to stay in the pool and jeer at them! You can rent a bathing cap, but you can also try to get one of those flimsy see-through shower caps that some hotels provide among their amenities. That's what I used, and although it looked kind of silly it worked just fine and took up basically no space in my luggage. (Thanks to the RS book for the excellent suggestion!)