Budapest advice - ideas of what to do

[TrepidationOfTheSpheres]

I posted this in Chat last week, and got a few helpful replies but threads move too fast there, and I realised that this would be a better place to post to get advice. So I'm reposting here for any further thoughts! Keen to have any particular recommendations of tours/guides etc that might need booking - there are loads available and mixed reviews, so any personal views welcome.

I'm going to Budapest in half term, and am starting to look into what to do. (two adults)
If you'd got any suggestions for me, I'd like to know

1. Castle area. We are staying near here. Fisherman's Bastion looks very pretty, and free to wander around. Also Matthias Church. I am not sure which of the various other things to do - there seems to be various museums, halls, etc, and no one site to buy tickets to it all, so needs to make choices I think. The hospital in the rock sounds interesting. What else is there and which museums would you recommend? Is there actually anything of the 'castle' to see, or is it all museums now? Would you recommend a guided tour of some/all/any of it? I have heard about caves too - some in the castle area and maybe some elsewhere? We could probably spend a whole afternoon or more here, given that we're staying kind of near, so could also do bits at different times.
2. We'd like to see the market and maybe do something like a food tour, food workshop, wine tasting, or similar. Not super expensive though. Any recommendations?
3. Would you do a cruise on the Danube? If so, any recommendations of companies, or is it mostly one company that does them?

Other things that we were planning are one of the baths (I guess Széchenyi is probably most likely as the Gellert one is closed, though could maybe do others as well?), Museum of Terror, maybe a Ruin bar, some of the other sights we can see just by walking around like the Shoes on the Danube, Chain bridge, etc. Parliament? Ferris wheel? Margaret Island? Jewish area walk/tour? New York cafe? Caves? Hop-on-hop-off bus? Walking tour? (either of those could be the day we arrive to get our bearings). All of that could be decided when we're there depending how we feel, time, weather etc., so I don't need to plan a detailed itinerary or anything, just getting a few ideas of what kind of things are in similar areas and would be good to do on the same day (and checking in case things are closed on certain days). There seem to be various other quirky museums too, if we have spare time. We're on a bit of a budget so planning lots of walking and free things, but do want to enjoy it and do the main things. Breakfast included at hotel, and I"m sure we can have lunch easily, but will need dinners.
Comments on any of the plans so far? Thanks!!

Sorry this is long but hopefully helpful. DH and I went in Nov 2 years ago and loved it. We stayed in the Jewish quarter and walked everywhere.

-If you want to see the parliament, book now. We didn't consider booking till we arrived and it was booked out!
-We went to Gellert baths and spent the best part of a day there and really enjoyed it. Széchenyi is further away but larger. When I researched, they don't rent towels but will sell them to you. We bought sports, microfibre towels from either argos or sports direct and took them with us. I can't find a link to the actual ones now. We also took out own flip flops. I wrongly thought we needed swimming caps in ALL pools, but at gellert, it was only needed if doing laps in 1, specific pool. So we carried them around all day and didn't use them, so check on the pool website.
-Goulash is traditionally a soup, not the thick stew I thought it was.
-We visited the central market hall. It did feel very geared for tourists rather than a real, local market. The ground floor sold meat, fruit/veg, sweets and the upper floor had a few cafes and lots of tourist souvenirs. I think the basement had an aldi or lidl. I bought paprika from a local off licence and it ws cheaper and seemed more traditional than what was being sold at the market.
-We visited St Stephan's basilica. 2 ticket options. The main church or one which also included the view from the platform at the top. IMO, the church is far more interesting and unless you like stairs and/or queues, I wouldn't bother with the viewing option. There is a lift, but only for part of the trip, so you still need to be fit enough to climb several flights of stairs before and after the lift. Also, the lift only took 4 people, so there was a massive queue to get it.

There is a travel man episode which you might find on youtube. Have a great time.

The labyrinth/caves were so cool, 100% recommend. It was my favourite thing.

Monument park is interesting, you need to take a metro type train there.

Dont bother with thr market hall but a short cruise up the danube is a great way to see the city.

The fort at the top of the hill is not worth the climb!

Széchenyi was good, you need bathing caps and you will need to take a little metro type train there also.

Up near matthias church there were a couple of short displays to go and see,

Thank you both!

The Gellert baths are closed. I did read about needing a towel, though was hoping to take one from the hotel...! I have read to bring flip flops though.

Is the parliament worth seeing? I can book beforehand if it sounds worth it.

St Stephens basilica also sounds good, will add that to the list.

Which labyrinths/caves did you see? Are they part of the castle complex? I still haven't quite worked out what exactly is at the castle - it mostly seems to be a collection of separate museums I think? no actual castle?

I like the idea of Danube cruise - any companies in particular? Is there just one that seems to do all the dinner/cocktail/sunset cruises, or are there loads of different ones?

Good to know the market hall isn't that great; I'd wondered about booking some kind of food tour/workshop etc, but might just be quite touristy

Also, what would you do on the same day as a parliament tour? location/timing-wise.

Maybe a boat trip would be good for the first night. The budapest card apparently includes one or a discount or something, would have to check into that, but not sure it's worth it overall.

AI suggests that booking the hospital in the rock might also need to be done in advance.

I wasn't going to plan too much until I know what the weather will be, but I could have some rough plans. We'll have three full days and a couple of part days.