In the moderate category, I recommend Firkasz (Tatra utca 18), Fészek (Kertész u. 36), Le Jardin de Paris (Fo u. 20),
Apostolok (Kígyó u. 4; request the Judas table

), and the excellent game restaurant Aranyxzarvas (Szarvas ter 1, in Buda).
For a splurge, visit Gundel (Alla tkerti u. 2) -- not cheap but an excellent value. I highly recommend the game menu.
Hungary has a strong wine culture, so, as a result (arguably), its beers aren’t extraordinary. Dreher and Borsodi are the two main Hungarian breweries, and both make dark beers. You’ll find lots of Czech and other European imports. If you run across dark, spicy Kozel (Czech), order it.
Be sure to hit several cafés.