Lived in Budapest for a few years. No need to hire a car the public transport is excellent, get a 72h pass from a machine and download the BudapestGO app for routes and live schedules. Ride the number 2 tram all the way along pest side for great views. Bolt app is like Uber and works well in the city.
Hungarian wine is fantastic, try to book a tasting to know where to start. Doblo wine bar or drop shop will also be great for informal advice. If you ever go in September the wine festival at the castle is fabulous.
It can get very cold but not necessarily any worse than UK every year.
The usual tourist guide book stuff can easily fill up your short trip and is great fun. Fisherman's bastion is stunning and as someone mentioned already there is usually an ice rink close to heroes square in winter. You won't see the fountain on Margaret island as it only runs May-Oct and is covered over in winter as it would freeze. Visit the Basilica and climb it too for great views of Pest.
The thermal baths are wonderful in winter. I don't agree with PP who suggested Palatinus on the island - it's a strand/outdoor fun pool with a couple of thermal pools but not a fürdo and most of the outdoor parts are drained for winter.
All of the old thermal baths are a bit smelly and grubby because of the waters and their age so you have to accept that and they are not to be missed. Széchenyi is very touristy but the building still makes it worth seeing. I prefer Gellért but it was closed for refurbishment last year so do check if it's reopened. Rudas is nice but only one tiny, albeit with dramatic location, outdoor hot tub. Smaller with more locals than tourists, try Lukács or Király baths in Buda. ("keer-eye", also the name of the main tourist street in Pest where you will find all the stag do bars - some nice places hidden away in that area - just avoid at weekend nights).
It's a fabulous city for tourists. Eating out I love Két Szerecsen, Klassz, Leila's, Dobrumba, Mazel Tov.
Have a great trip, I can't wait for my next one after writing this!