Budapest is one of my favorite cities, too. I’ve been a few times, including a few days alone during one trip. It’s an easy place to wander and take in the sites without doing a tour. If you are wanting to do a tour because you want to be part of a group, that’s different. Your hotel could probably recommend a walking tour or you could find one on TripAdvisor.
For a self-made tour-the VoiceMap app is great for audio tours—I just looked and there are several tours in Budapest of places I’d recommend visiting.
I would definitely go to the baths at the Gelert Hotel. If you enjoy wine I’d go to a tasting in the Buda Castle Caves.
If you really want to do a day trip out of the city, I’d do one of the trips to Lake Balaton. There are companies that do bus trips that include a little boat cruise on the lake. Have fun!