Somewhat unique to Budapest and a must do on any visit are the extensive public baths.
The largest one (and most popular) is the Szechenyi Baths (built in 19915) with many indoor pools and 3 large outdoor pools set in a palace like building near the City Park. It does fill up at times in the summer so consider booking ahead.
Rudas baths is a medieval Turkish bath with a rooftop panoramic pool (check hours, days sometimes coed, sometimes single sex.
Gellert Baths is a beautiful Art Nouveau thermal bath, built in 1918 but closed for renovations through 10/2028