If you want to eat cheap, warm and good traditional food try FOZELEKFALO. They offer soup like - but thicker - vegetable dishes (fozelek) and if you want you can have something on the top (fried cheese/vegetables, sausage, fish etc.) It's becoming a chain now, so you can find them at many various locations in Budapest, but the first one that opened is in 18 Nagymezo utca (Andrassy corner, near Oktogon). This one is closed on Sunday but during the week it's open till about 9 pm. Be prepared that in meal-times there can be a long queue but it's going very fast. You can also ask for take-away.
Even if the prices are a bit highert in Budapest than in the rest of the country, you can eat for cheap and traditional dishes at good restaurants all around town§country!
Drinks are quite cheap aswell, that's why I received many complains related to "drink tourism" (people who only visit hungary and eastern european countries to drink and make pub crawl!).
I am sure that culture and good concerts can keep you away from this kind of behaviour!
Try Palinka, national water of life, most of the beers you'll find are Tchek...