The bath isn’t a bad way to spend an evening. The shower cap isnt required in the main pool, but the others, yes. Some enjoy it a lot, others not so much. I do like them, but if I only had two nights and a day I might do something different. But that doesn’t make it a bad idea.
The one that will be easiest for you will be http://www.szechenyibath.hu/ which will set you back about 6.000 HUF a ticket (with a cabin, a but cheaper if you change in the locker room or go after 7pm) or roughly $20. So, not real cheap. Easy to reach as it’s the next to last stop on the M1 metro.
You didn’t say where you were staying so its hard to suggest a plan for the day.
Using the 2 Tram you can see the:
Corso (Danube Promenade)
The 47/49 Tram will get you to the Dohany (Great) Synagogue and the old Jewish Quarter as well as near the end of Andrassy ut.
The M1 which runs under Andrassy ut will get you to the Opera House and the bath house. But then walk back down it, as its worth the walk. The Opera is covered up for renovations but they still run tours most days. Look for the sign in the front.
Gellert Hill, best done with a taxi and a bottle of wine after sunset.
Margaret Island you reach on the 4/6 Tram. Not sure I would spend a lot of time with that. Too much else to see.
Buda you reach, if you are staying in Pest, by crossing the Chain Bridge and taking the funicular up, then walk down. The funicular is 1,000 HUF to go up (about $3.30) or 1.700 HUF round trip.
So, the circle is Andrassy ut on the 47/49 to the Synagogue, then on the 47/49 to the Great Market Hall (ooops, closed on Sunday), then on to river where you get on the 2 Tram to the Chain Bridge where you walk to Buda, then down the hill to the 19/41 Tram which you take to Margit híd where you get the 4/6 Tram to Margaret Island, then back on the same tram to the Pest side of the river where you can get back on the 2 Tram or to Andrassy ut and the M1 Metro. Get a map.
Don’t know anything about boat parties.