Design, typography, illustration, animation, art, comics, graphic design in Budapest?

When I was in Budapest in 2008, the Museum of Trade and Tourism (then known as the Museum of Commerce and Catering)
had two exhibits. On the ground floor was "Food of the Renaissance, and The Renaissance of Food" (the latter referring to after the fall of communism), and on the upper floor was an exhibit of Hungarian recordings. Both of these had lots of vintage illustrations, ads, etc. and would have been right up your alley (the recordings spanned many decades, and some of the sheet music, posters, etc., were quite elaborate).

Here's the website; you can see what's on view when you will be there: http://www.mkvm.hu/indexen.php

By the way - great question! This is exactly the kind of thing a forum is so useful for, as this information is hard to get from guidebooks.

Brody Art Yard: Here is a place that you might find interesting. Opened up last year in the building adjacent to the apartment block where we stay when we are in town. Great little side street that connects the Opera House to the old Jewish Quarter. Interesting synagogue across the street and my favorite wine bar on the corner of Kiraly utca about 200 feet from the front door.

https://www.facebook.com/brodyartyard
http://shop.brodyartyard.com/

I'd highly recommend the Museum of Broken Relationships in Zagreb. It's not focused on design, but the design of the "museum" itself (honestly, it feels more like an exhibition) is very interesting, and will likely appeal to any design lovers: https://brokenships.com/

Same thing with Budapest's House of Terror - the subject of the museum isn't design, but the design of the museum is so integral to the viewer experience. Probably the best-designed museum I've ever visited: http://www.terrorhaza.hu/

Happy travels!