Budapest is an unexpected and very charming mix: the feel of a soviet sanatorium, beautiful modernist architecture, and prices far lower than you'd expect in Europe. It's the full European experience for very little money and that wins you over immediately.
The big discovery is just how good and how cheap the food is. Goulash and paprikash are excellent literally everywhere you go. We tried lángos at Drum Cafe — the place itself is wonderfully authentic, and beyond the lángos there's plenty of other local cooking worth ordering. And kürtőskalács with mulled wine is best enjoyed up on Castle Hill, with a winter view over the city. In short: the number one thing to do in Budapest is eat :)
Beyond the food, a couple of things stuck with me. A huge part of the city historical center was destroyed in WWII and rebuilt authentically afterward — once you know that, you look at the buildings differently. And the old Budapest metro is its own charm: cute, atmospheric, and one of the oldest in Europe.
Honestly, I couldn't really find a downside. Everything is cheap, nothing felt like a tourist trap, and in terms of safety it's one of the most relaxed places I've been in Europe. The language is unlike anything familiar, but the locals are friendly and welcoming.
You can get around however you like - it's all convenient, but taxis are so cheap here that you can comfortably rely on them. Any season works, but winter has one special trump: sitting in an openair thermal bath while it snows, like those famous Japanese macaques.
Who I'd recommend it to: pretty much everyone. It's that rare city that works equally well as a first taste of Europe, a food trip, a relaxed, low-cost getaway or a resort town with thermal baths.