Budapest Tips:

Budapest is bigger than a majority of European cities but it is still walkable, just pack appropriate shoes!
The city is a little grungier than most, with a relatively large amount of homeless people and a general feel of not super clean, but don't fret, it's part of the charm!
It's a very "hip" place, not your traditional adventure.
Be prepared to eat extremely well. I had some of the best meals of my life here. Spinach and cheese "crepes", goat cheese and tabouli, food trucks, gourmet breakfasts, and so much more.
The city is at its best at night, be sure to go down to the river to see the castle lit up

Adventures & Ales Top Recommendations:

Do the free walking tour. You'll see a ton of the best spots all in a few hours with all of the local history and secrets. Much better here than in most other cities!
Check out the ruins bars, most notably Szimpla Kert, and then go to Karavan for street food next door afterwards, nomsssss.
Eat breakfast at New York Cafe. My goodness I cannot recommend it enough. I ate all of the food pictured above in this gorgeous setting
Climb Gellert Hill to the Freedom Statue for the best views possible of Buda + Pest + the river
House of Terror and Berlin Wall Memorial: On Andrassy Avenue, the museum is informative but engaging and the memorial outside is very cool to see.

The Classic Tourist Spots Actually Worth Visiting:

Walk the Chain Bridge! It is one of a few bridges that connect Buda and Pest and is gorgeous, the lions are pretty sweet.
Go up to the Buda Castle for great views of Pest and the river
Matthias Church and Fischerman's Wharf near the Buda Castle are both gorgeous, definitely check them out while you're up on the hill.
Go to Danube Promenade to see the shoes memorial then keep walking to the Parliament Building, an incredible architectural feat

Other ideas:

Wander the Jewish Quarter: The Great Synagogue, the Holocaust Memorial Tree, Gozsdu udvar, and the Orthodox Synagogue are worth checking out
Spiler Bar in the Jewish Quarter
Akvarium Klub: Hang out outside on the water with the locals, it's the perfect place to just chill during sunset, pictured above (the inside is lame)
District VIII: Wander the streets, end up at The Tasting Table for cheese and wine!
Heroes Square and the City Park--home to one of the most famous baths of the city
The Central Market: Tons of pop-ups: clothing, souvenirs, fruit, veggies, meat, all sorts of things