First of all, I'd do the various free tours which take place multiple times daily (many cities have these and I've almost always have wonderful experiences.) given by various companies. Google free tours budapest for info.

If it wasn't the middle of the winter, I'd suggest renting bikes or doing a bike tour.

I would highly recommend going up to the citadel like an hour before sunset and staying there until it gets dark, for a spectacular and enchanting (and romantic) view.

Make sure to stroll down Andrassy ut. At the end, which you might need to hop on a bus or cab to as it's 2.6 km, you'll see Hero's Square, Varosliget Castle, Szechenyi Baths and the zoo.

Check out the parliment building, as well as the "Shoes" Holocaust memorial outside the Parliment on the river bank.

Nearby, there's a larger than life statue of Ronald Reagan.

In the Jewish quarter (covered by free tours which are highly recommended) check out Szimpla Ruin Pub. Very creative and interesting as well ad fun.

Check out Nagy Vasarcsarnok market.

For a fun experience a bit outside the center city, visit Celeritas shooting range. The let you shoot a wide variety of weapons including many soviet models. Ask them to help you arrange a cab ride there and back with other customers.

If you like opera, watch one at the opera house.

To get from the airport to the center of town, some airlines (like Wizz) have shuttles. You can ask the hostel or hotel you're staying at to arrange a taxi to meet you. Ask them what it's supposed to cost and don't let the cab driver rip you off as is commin there.

Enjoy your trip!