We went in Dec. I hadn't been to Europe since covid. Many of the sights were half price for EU citizens and full price for other countries, including Brits! If you have a European passport- take it with you!
If you want to see the parliament- book it now! We went for 5 days and it was already fully booked!
We went to Gellert and spent half the day there. We'd wander between the outdoor, heated pool, then indoor ones, the steam room, sauna, then start the round again! A few hours though would be lovely too. Note, that we felt absolutely drained and knackered afterwards. Not that we did anything, other than walk from pool to pool, but I think the heat made us tired. Its only across the bridge from the food market. That was worth a quick look. Food down stairs and typical, touristy things upstairs. I ended up buying paprika elsewhere though.
The Basilica. The tickets are from a shop on the street nearby, not at the actual basilica door. 1 ticket is just inside the basilica, and a combo also includes going up to the top look out. If short of time- I'd just see inside. There were 2 lifts to the top, but one had very long queues, because the lift only held 4 people at once. We waited maybe 15mins to get up. You then still needed to climb up more stairs at the very top. Yes, the view was nice, but we spent 5 mins looking out at buildings which we had little idea what they were, then needed to queue again to get the lift down again. Yes, you can walk the entire height up and down stairs, but I had a knee injury.
Have a fabulous time.