Friend of mine has lived there for about 18 years.
He says one bit of advice he has for tourists is that all the usual scams are present, like tourist prices in restaurants, snatch-theft, fake merchandise, fake beggars, rigged taxis, fake ATMs and currency exchange scams, but tend to be a bit more prevalent in Hungary and scammers can get quite pushy about it.
But the main thing to watch out for is the light-fingered Gypsy types, especially the children - They'll swarm you and you'll have twenty small hands in your pockets all at once, and are so persistent to the point where you have to beat them off, quite literally. The cops won't even bat an eyelid if they see you smacking a bunch of kids away. If you're having difficulty, and they can be bothered, they might step in to lend you a hand.
However, the traffic cops (who are armed) WILL take a dim view of speeding... more so if you're doing over 100mph on one wheel as you go past them, as the aforementioned friend found out (and is why he doesn't have a Fireblade any more).