Well, Budapest is really all about the vibe, so you can't go wrong just wandering around and checking out things that seem interesting. Some of the sites that I really enjoyed, though, were the Great Market, which is worth a look. I didn't spend a ton of time there, but it's a fun place to visit. And St. Stephen's Basilica is gorgeous and has many interesting pieces of art inside, including a reliquary that holds St. Stephen's hand.
I highly recommend visiting the Buda side, especially Matthias Church, which is one of the loveliest churches I've ever been in. It was just beautiful, and you have the benefit of being up high so you can look out over the city with an incredible view.
I didn't make it there, but Hospital on the Rock is very popular with visitors, and I hope to get there next time. And of course, Parliament is an incredible building. My tour was canceled, so I didn't get to see it, but again, I hope to see that when I get back. It's a beautiful building from the outside.
If you want to relax a bit, visit one of the thermal baths. And if you like music, it's worth checking out what's playing at the Hungarian State Opera. I saw Turandot when I was there, and it was one of the highlights of my trip.
Here's a link to my trip report if you'd like to read it. It has some lists of good restaurants I enjoyed that might come in handy. https://community.ricksteves.com/travel-forum/hungary/budapest-mini-trip-report-the-opera-the-scenery-the-food-the-catastrophe
Edit: kmkwoo above mentioned that trams are free for those 65 or older, which is true. In fact, all public transportation is free, including the airport bus, trams, buses, and trains. You don't need a ticket, but just make sure you have your passport with you in case you're asked for proof of age.
