Gellért Thermal Bath is on the Buda side next to Gellert Hill, which can be good to climb for a nice view of the city. The street that leads up to it, Bartók Béla út, has hip cafes and bars, like my friend’s vietnamese Caphe by Hai Nam, so can eat/drink around there.
Crossing the river to the Pest side from Gellért Thermal Bath, you have the Central Market Hall, which is a nice market and you can walk from there on Váci utca, to Vörösmarty tér which is a restaurant/shopping street with nice buildings and can also go down to the riverside.
Also around there is the Jewish corner, with cool pubs and nice to walk around, like the most famous ruin pub Szimpla Kert, which is touristy, but still definitley worth a drink/visit, some restaurants like Gettó Gulyás, which serves hungarian food and places around Gozsdu Court.
Also around there you have St. Stephen’s Basilica that has wine bars and restaurants around (although expensive). For hungarian food Retek Bisztro looks nice.