Just got back from a lovely week in Budapest and wanted to share my experiences since there seemed to be a bit of a dearth of information.
Our first dinner was at Konyha which my sister picked because it is right around the corner from our hotel and I'd just gotten in from 16 hours of travel. It turned out to be a lovely restaurant and we returned for breakfast another day. It's near Deak Ferenc, which is a main public transit hub on the Pest side of the river and turned out to be an excellent location overall. The restaurant is simple, very friendly English-speaking staff, and they do make a real effort to source fresh local ingredients. Their whole trout stuffed with lemons and herbs, grilled, and served over a bed of arugula, was excellent. The rest of my party was vegetarian and we weren't drinking but our bill still came out to about US $12 each for dinner and a few shared salads and soups. Amazing. Breakfast a few days later was really excellent- a nice chewy bagel with poached egg and a sort of a creamed spinach/pesto hybrid topping. Kept me completely filled up all day for about $2.50 US.
Other breakfasts were not as successful. We loved the coffee at Bluebird Cafe and Coffee Roastery (in the Gozsdu plaza area, a giant mass of mostly bars and restaurants), but their pastries and breakfasts were pretty awful. Get an americano to go and pickup food elsewhere. Also, in what just strikes me as weird, they sprinkle just about everything you order with peanuts. This would be a giant no-no in the US due to allergy concerns. I really did not expect my cinnamon bun to be slathered in sauce and peanuts when it came to the table, and when I asked if those were in fact peanuts on top, they first denied it, then confirmed it, and never offered to take it off the bill or get me anything else. (I'm not allergic; I just don't like unexpected peanuts) The next day, we went back because the coffee was just such a good antidote for jet lag and I was served a small plate of pancakes... covered in peanuts.
For lunch, we fell into the tourist trap of Fashion street once. It's hard to pass through and not eat. I enjoyed a plate of sausage and stuffed cabage, but compared to food everywhere else, it was quite expensive.
However, do NOT miss the Kürtőskalács "chimney cake" pastries in the fashion street tourist area. They are fantastic and unlike anything else I've had. I tried to convince a vendor to move to the US, assuring him he'd become rich, but no deal. They are rolled sweet yeasted dough which is wrapped around a special wooden dowel then roasted on a spit over coals. The layer of sugar on the outside caramelizes and the dough on the inside remains soft and puffy. Please, someone come make them in Boston.
Our one fancy meal ended up being lunch at Borokhonyha, also near Deac Ferenc and which recently received a very good write up in the NYT. Probably owing to that, they were unable to take us for dinner. No matter, lunch was fantastic. When they heard us discussing the term "pike perch", the waiter brought over a fish atlas and gave us a very nice zoologic explanation. This is the kind of service that gets you a Michelin star. I would highly recommend making a reservation and having a meal there because although it's a fancy restaurant by Budapest standards, the prices are still very reasonable overall so this is your chance to experience a high level of dining on a budget.
A few other meals- a nice dinner at Trattoria Pomo D'Oro. Very old-world Italian feel. I had an excellent duck breast and the vegetarian members of our party were happy with their flatbreads and pasta. Even with a few glasses of wine, our party of 4 had dinner for $60 US.
We had one dinner at Pater Marcus, a Belgian Beer pub on the Buda side of the river near the castle. Let's just say they didn't convince me they'd been to Belgium. They only have about 10 beers on tap, trending heavily toward very fruity selections, and the food was really not very good. I got off flavors from some mussels, and the "frites" were mushy wedge fries of the Sizzler Steakhouse circa 1980 type (a lousy American chain steakhouse for those wondering what I'm talking about). Budapest seems to have a pretty good wine culture going, but if this was their venture into beer they're going to have to step it up a notch before I'm impressed.
We had an interesting Georgian dinner at Hachapuri- I was completely unfamiliar with Georgian food, and would love to explore more. I liked the ambiance at the restaurant and we were able to order by pointing at pictures on the menu which had good English descriptions. This was good, because the staff didn't speak much English. Dessert after that was a trip to the Bookstore Cafe, an architectural gem of a mirrored, gilded ballroom hidden upstairs in a bookstore. Again, stale awful pastries and cakes. Have a cup of tea and then get some gelato if you must, but worth it for the decor and wow factor of the space.
That about sums up the eating. Oh- if you're there on a Sunday, do try to visit the Farmer's Market which is located inside the Ruin Bars not far from Deac Ferenc. We had to have someone help us find the entrance but it's pretty amazing once you're in there. Lunch of fresh yogurt, sausage and cheeses is a nice break from restaurant meals and you can find lots of nice food gifts which are a cut above the typical tourist bag of paprika.