Cool! Ok, not sure which way your tastes run so her is a grab bag of options.
Just outside the city:
Take a day trip to the Danube Bend. Starting just north of Budapest (like 20 minutes by car) is Szentendre, a popular medieval town. From there continue to Visegrad(another 30 minutes or so) and visit the castleatop the hill - amazing views of the Danube bend from up there. Then go to Esztergom(another 30 minutes) and visit the Basilicathere. Then cross the bridge into Sturovo, Slovakia and check that off your list of countries you’ve set foot in! As well, have your cameras ready for the drive back over the bridge because there are dramatic views of the Basilica.
Another short day trip is to Godolloto visit the Royal Palace there. The town is an outer suburb of Budapest, so not far at all by car. And once you’re on that side of town, another really cool thing to see is just another 1/2 hour up the road: the bear and wolf sanctuary at Veresegyhaz! It’s open every day from 8:00 to 7:00 and you can feed the bears honey on a long wooden spoon. Everyone we’ve taken there has gotten a huge kick out of it - highly recommended! (Note - the signage to find the place is not great, but “medve” is the Hungarian word for bear if you need to ask directions.)
Inside the city:
Go to the top of the Gellert Hill - best views over the Danube. Up there you will find the Liberty Statue and the Citadella, a fortress that saw heavy action during WWII and the Hungarian Revolution.
At the base of the Gellert Hill is the Gellert Hotel and Spa, an excellent spa and worth a visit. Even better are the grand Szechenyi Baths in the City Park, not far from the Heroes Square. You can go there even if it’s cold out. The baths are thermal. We have even been there while it was snowing!
At the north end of the Castle Hill is the Museum of Military History and just outside the city proper on the southwest is the Memento Park where you can check out some of the massive statues from the Communist era.
On the northern end of town on the Buda side are the ruins of Aquincum, which while interesting are not especially impressive. If you want to do a quick and impressive Roman ruin, go see the Military Amphitheatre in Obuda (part of Budapest).
I never shot guns in Budapest but we did visit this placeoccasionally to practice archery. They have a small shooting range in the basement, and you can rent gear and shoot for an hour for the equivalent of about $8.
Food:
Two words - Raday utca (Raday Street). 30-40 restaurants in a row. The street we lived on. It starts at Kalvin Square (a major subway stop) so very easy to get to. You can find any kind of food you like, from cheap and cheerful to Michelin rated. If you want authentic Hungarian food, the Vorospostakocsi(Red Mail Coach) has it, but it’s pricey. The best food on the street is at the Manga Cowboy, although it’s East/West fusion stuff and not Hungarian at all. And if you just want an overwhelming foodstravaganza experience, wander through the Nagy Vasarcsarnok (Central Market Hall) any day until about 3:00 PM except Sunday - it’s big, beautiful and bountiful. Worth a visit. It’s also close to Kalvin Square, closer to the Danube, and the southern terminus of the main tourist shopping street, Vaci Utca (worth a stroll but overpriced).
Liszt Ferenc Ter (Franz Liszt Square) is also good for food but it’s a bit more touristy and a lot more pricey. But it’s along Andrassy Ut, which terminates at Heroes Square.
Ok, plenty there to get you started. I’m off to work but will no doubt think of other stuff so will inundate you more later
From the above list, I’d recommend the Danube bend, the bears, the Central Market Hall and Raday Utca to get on your list for sure.
Cheers,
aesop