What part of Budapest, as a base?

If it's 3 nights or more I like along Andrassy ut (District VI) for all that you mentioned. 3 nights or less I think along the river in District V makes sense. To avoid? To avoid the worst of the tourists, avoid District V, to avoid endless travel time to the majority of thr sites and things to do, avoid Buda. To be safe .... well, I can't imagine a location you might consider that is not safe.

EDIT, unless you are into the party scene, avoid District VII.

How do you know? By postal codes, like "1061"
The fist number, 1, is Budapest
The 06 is District VI
The last digit is a part of the district.

Do some searching and post what interests you and myself and others can probably tell you about the hotel or the neighborhood.

I would recommend the districts along the Danube, especially districts 1, 2 and 5, from where you can easily get to the sights on foot or by tram.

For a first time visit, the Pest side (east side of the Danube) is the most conveniet location. The airport and major train stations are in Pest. Budapest's public transportation system is fantastic. It is inexpensive, easy to navigate, and will get you everwhere you want to go. If you can stay close to a Metro station or tram stop, you'll be able to maximize your visit.

Mister E's suggestion of a hotel along Andrássy út is a good one. Check out Hotel Moments: https://hotelmomentsbudapest.hu/ or other hotels in that area.

Christi I keep walking past the Thai Hotel on Andrassy ut and finally looked it up. Worth looking at, then if someone stays there, let us know how it was.

Where to stay is really a personal thing; what you are interested in, what you will do in the evenings, how fancy or how basic your taste. District I is part of Buda and is where the Castle Hill is located. I think its a lovely place with amazing views of Pest across tge river, but so far removed from so much of what there is to see and do. District II is also in Buda and the only thing that sticks with me about District II is that's where I had a colonoscopy a few years ago; and not terribly convenient to much; but again a spectacular view of Pest.

Another interesting place you might consider is the lower end of District VIII. I dont know anything about the hotels in the area, but its a lovely and interesting zone and if you stay close to the 47/49 tram line, it could be a nice camp.

Two weeks ago we stayed at the Hotel Moments for 3 nights. We felt the hotel was a great value and in an outstanding location.