Old mineral baths with old men playing chess. Ruin bars. A split city.
Buda was one city on the east side of the Danube River and Pest was the city on the west side. They merged in 1873 and became Budapest. I didn't get a lot of nosh or nip photos but have a few recommendations for where I went.

Nav:

City Walk: Finding a guided or self-guided tour is a great way to see a lot in a little amount of time.

Szechenyi Baths...multiple temps, multiple mineral variations, indoor or outdoor.
Hungarian Parliament Building
Shoes on the Danube...a tribute to the 800 Jews shot on the edge of the Danube River by Nazis in 1944 and 1945.
Chain Bridge...the link between Buda and Pest
Fisherman's Bastion...
Looks like a castle or old fort...
It was built in 1902 as an art and tourist venue
Matthias Church
A hospital built into the mountain in the 1930s in preparation for WWII bombs.
Buda Castle...started in the 14th century.
Matthias Fountain...King Matthias on a hunting party. Located outside Buda Castle.
Danube River
St. Stephen's Basilica. Completed in 1905 and named after the first king of Hungary.

Nip:

Ruin Bars: These are a Budapest creation. Mostly located in the 7th District. The Jewish Quarter was largely abandoned by residents prior to and during WWII. As roofs collapsed and windows rotted, the shells of once beautiful building are all that remained. In 2010, a trend began of stringing up lights and setting up bars in the courtyards. Szimpla Kert was the first and more followed. Some ruin bars are so renovated they aren't ruins anymore and some are more...rustic. Find one that appeals to you and go. I went to Anker't because of the open air / no window vibe. Unfortunately it's permanently closed.

Unicum: An digestif containing alcohol. Reminds me of Underberg from Germany. It's an herbal tincture that was created to help the Austro-Hungarian court in the 1790s with digestion. If you've had Jagermeister, you have an idea of what this tastes like.