The two most famous are Széchenyi https://www.szechenyibath.hu/ and Gellert, but Gellert is closed for renovation. Széchenyi can be a bit touristy, but still very much a hangout for the locals in Pest. Sometimes is gets crowded so mornings are maybe better. Pretty grand architecture as well.
If you want to go very local, then Lukács is a good choice. Nothing but locals and expats. https://www.lukacsfurdo.hu/en/. Not grand like Széchenyi but all the little rooms and stairs up and down and the outside pool in the courtyard are fun.
Rudas contains one bath that dates from the Ottoman occupation in the 16th century. Pretty impressive architecture, but most of the facility is fairly new. Not all of the facility is co-ed on all days. Do check the web page. https://tickets.rudasfurdo.hu/partner/rudas I have never been so I can’t help beyond that.
Veli Bey is another 16th century bath house but I know nothing about it. Its privately run and operated. http://www.irgalmasrend.hu/site/velibej/home The other two Ottoman baths, Kiraly and Vac are closed for renovation.
Then there is a whole collection of other, modern, baths that are popular with the tourists. Most of them are operated by the same agency so they are located here: https://www.spasbudapest.com/
Back to your question, I would suggest first Széchenyi which you can make part of a visit to the park, House of Music, Heroes’ Square, Lake, Vajdahunyad Castle, Museum of Ethnography, maybe Városligeti fasor, then since you are here for so long, do a local thing at Lukács.
With just five days, unless you have been there before I would say a day trip to Szentendere and maybe an overnight in Eger. But you could just as easily stay in town for 5 days and still have things to do; especially on a first visit.