Any local recommendations for the capital city of Hungary?

Hii,
Me and my boyfriend met in the capital city of Hungary and now we are going back together 3 years after. We are super excited, just to relive and reexplore this place now together.

Therefore, we’ve been talking a lot about it and we would love to live like locals for the time we are there.

Anyone that lives/has lived in there and/or are hungarian, please give us your insights on what are the best places to eat (we are both vegetarian), neighbourhoods to explore, romantic places, where to have a drink, what drinks should we get, anything that you think is typical for the capital city of Hungary:)

We would highly appreciate it!