Headed there at the end of the month.
We were there last summer. Get a transit pass, it’s easy to get around even if you don’t have the language. Take a day and go to the Gellert Baths. Buy tix online in advance. It’s around $20/pp. Bring your own sandals/flip-flops, and towel (they charge extra if you use the ones on-site). Glorious recently-renovated classic spa. Everyone goes to the Szechenyi but we spent all day at the Gellert.
Outdoors in winter is great.
Went a few years back and loved it, but did everything back to front and only got bearings a couple of days after arriving. Would recommend getting a travel pass that gives you access to the metro and local trains, its more spread out than i had anticipated so walking everywhere was a bit too much (i tend to hike everywhere so was surprised when caught out). The parliament building tour was/is free but you had to book a slot online in advance and was about an hour of potted Hungarian history. The castle complex provides wide panoramas of the city and surroundings, and is accessed by a cheap funicular railway. The historic market is great, but can be pricey depending on what you go for. There is an archaelogical Román site/outdoor museum at Aquincum which is a few stops Up on the riverside rail line from by the market (we stayed not far from the aforementioned Gellert baths). Loads of places to eat & drink but we found where we stayed (west of Danube) that restaurantes & cafés were more readily available at lunch rather than evening as locals tend to have evening meals at home. There were loads of bars/places to go at night, but will doubtless have changes in past few years