First of all try to get the "Index", a monthly gallery guide that includes almost all interesting places of art. Opening times are usually Tuesday to Saturday. You can check their programme on the Internet, those are the ones that I found worth visiting:
Studio galleryArtist run space and young artist's association. Offers studios and residencies also and has an interesting programme as well als publications.
(map)Trafo galleryGallery in the Trafo cultural center. Seems to have a focus on political/critical art, offers a very interesting perspective though.
(map)
Kitchen Budapestvery interesting gallery/lab for digital art. Organizes workshops and lectures and offers positions for researchers in this field.
(map)Videospace Budapest(map)
Mucsarnok/Kunsthallebiggest institution for contemporary art. Located next to fine arts museum, quite interesting programme.
(map)Ludwig MuzeumHuge space sponsored by the Ludwig Foundation (Germany). Various temporary exhibtions, seemed a bit commercial to me when I was there.
Lumen Photo GalleryDon't go there only to see the exhibiton cause it's really small, but it is interesting and a very nice place to take a coffee.