Many people don’t bother to look up at the top of buildings, missing much. Aside from the historic listings, Magyar Államkincstár on Hold utca across from where Perczel Mór utca comes into it is one of the favorite buildings in Budapest; it is now part of the National Bank. Move around so the trees don’t obstruct your view.
Párizsi udvar in the Art Nouveau style, where Kigyó utca meets Ferenciek tere, is sumptuous both outside and inside. The building has been sold, so it may not last forever. See it while you can.
Walk in the courtyard and look at the ceiling. On the körút, near Blaha Lujza tér, is the New York Palace built in eclectic style with an emphasis on Italian renaissance and baroque.
For something more modern, don’t pass up the Lehel tér market at the Blue metro stop (line M3) by the same name. It looks like a beached boat.