In Budapest, you can pay by card almost anywhere except the tinyest shops, and in most cases, you can do it contactless.
In tourist areas, some shops take only cash and try to profit from ridiculous conversion rates if you pay by €. There are some tourist-focused restaurants too that don't accept cards (IMHO a bit suspicious), as George said, look for signs or logos on/near the entrance. Museums should be fine.
Having a minimum amount for card payment is not allowed, but a few places still do it, usually with a relatively low limit (e.g. 1000 HUF).
As for public transport: you can buy tickets/passes by card at customer service points and vending machines. Note that while it is officially possible to buy tickets on trams and buses from the driver, it's by cash only and they really don't like it when you try (takes time, they have no change etc.) so try buy them in advance.
(source: I live in Budapest)