First everyone goes or maybe stays at the Buda/Castle side and explores the old city and enjoys great views over the river, Parliament and new city. We stayed at that side at the Hilton, we had a river view room and it was gorgeous each night with the City lit up below. If you're not staying on that side, you can either walk up (we did once going back to the hotel, it took a while but was good exercise), catch the bus, taxi or better yet, ride the funicular up to the palace on the top of the hill. Then wander around.
When we went in February the weather was close to freezing but enjoying an outdoor thermal bath such as the Sczechenyi Baths is a must. I don't know how it is like in summer/autumn but when its 1 degrees outside but 30 degrees in the water, it was really refreshing! Budapest is renowned for its many thermal baths.
Go to the Central Markets. Its pretty tourist-y that sells the usual market stuff eg fruit, veg's, meats, souveniers but what my tip would be is, there are many food stalls on the 1st level where you mix it up with the locals, enjoy hungarian food whilst enjoying the hustle and bustle of the main market floor below.
Get to the markets by wandering along Andrassy Avenue. Its a famed pedestrian street with lots of shops but explore off that avenue to see interesting shops.
There's other historical things to see / do but I am not a historical kind of person so can't give you much more info on that

I am more of a geographically kind of person

That is, been there done that, tick the box type

Enjoy your time there!