I used to live there for a few years. Can recommend the Rudas bath house. It's authentic and nice for some hours. I much prefer it over the other larger ones.
Red bus to take you around is always how I would take visitors around, hop on and off, it sounds a bit touristy but you really just get to the right places.
Of course go check out the ruin bars and have a few drinks.
Good wise, it's changing a lot with my friends who still live there naming different places all the time...fisherman's bastion has a great restaurant in the evening which you can look down onto the city also. Touristy, but it's quite nice.
Do try the palinka. If anything more pops into my head, I'll let you know.
Ps. Taxis just walk outside to the kiosk and get one with a ticket for the driver. Lots of scams.
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