Thought this may be of interest. Attended a performance at BMC - Budapest Music Center, classical. Cost about 2800 Forints, under $10. Amazing concert. Also attended a performance at the Ferenc Liszt academy, also excellent.
Can't go wrong with either.
For what it's worth:
Website for BMC, bmc.hu
Website for Fer. Liszt academy, zeneakademia.hu/en/home
There are other classical music venues too. I really enjoyed both of these - BMC is more of an "in the know" venue, whereas the academy is more premier.
I also just saw further down in the posts about the outdoor classical or the other venues such as the Budapest castle. Definitely do those - Budapest in summer is supposed to be even better than in Spring (when concerts haven't yet spilled out into the open!). That said, the above venues, for what it's worth, were great.