DH and I went in Nov, 2 yrs ago and enjoyed it. I think we were there 4-5 days. We walked everywhere! Sorry this in long, but hopefully helpful.
-We thoroughly enjoyed the Gellert baths and spent half a day there. I don't think any of the baths rent towels/flip flops so you either need to buy them or take your own. We bought microfibre towels from sports direct or argos and took flip flops. You only need a swimming cap if doing laps in the main pool. For the steam rooms, sauna, heated and plunge pools- no cap needed.
-The houses of parliament tickets sell out quickly, so book before you go if interested. There were no tickets left when we arrived, so I can't comment on what its like.
-St Stephens basilica is worth a look for the beautiful architecture and to see the mummified hand of Hungary's first king 😬You can buy tickets to the main basilica or a combo ticket to see from the viewing platform at the top. IMO, don't bother with the viewing from the top. You either climb up multiple steps to see the view or get the lift. The lift only holds 4 people and was woefully slow with a massive queue. Out of the lift, you still need to walk up further flights of stairs. The view was ok, but unless you know Budapest, we were looking at random buildings and spent 5mins up there. But over an hour to get up there and back down!
-Shoes on the Danube. The pics you see show just the shoes and I imagined it as a quiet, sombre, place of reflection. Instead, there were burnt out candles, dead flowers, political flags and about 100 people taking selfies.
-Central Market hall. Across the bridge from Gellert spa. The bottom level was fresh fruit/veg, eggs, meat, herbs/spices and the 1st floor was touristy things. Some rather tacky and the typical tat you see in charity shops like a mug with Budapest written on the side. A few nicer things mixed in though. The basement had an aldi. TBH, I found a better choice and cheaper options for paprika in the aldi and a small corner shop than in the market!
-Buda side, the castle and fisherman bastian. We walked across the chain bridge to Buda. You can walk up the hill yourself to the sights or pay for the funicular- this had long queues though. Another option, and there were multiple touts around selling tickets, was an open aired mini bus. It drives to the top and claims to be a hop on/hop off bus. It is, but once a the top, we walked all over ourselves and only used the bus to get back down the hill again. I can't recall how much more it was than the funicular, but the queue to that moved quicker than I expected in the hindsight was likely the better option.
We stayed in the Jewish quarter and tried few restaurants near us. All very nice, traditional cuisine, but nothing I'd rave about. Do try a chimney cake and a langos (similar to a pizza). Goulash is traditionally a soup, not the thicker, stew I was expected. Since returning home, I've read good reviews of Spoon Restaurant which is a boat on the Danube. I'd return to Budapest, but then there are other places I'd also like to see. Have a fantastic time.