We've had good experience with the cards purchased in Europe. Not so good with the cards from home that we've taken with us although they are supposed to be international cards.
The Szechenyi baths are popular with locals and tourists alike. One of the metro(subway) lines runs past the opera house (worth consideration for a visit) and out to the baths. I seem to remember there is also a zoo in the area of the baths. I also remember that we were advised to leave our valuables locked in the hotel safe and take only the cash required for entry and metro fare. It was probably a good idea but we didn't have any problems.