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Get a tram pass and a map
download this app. the company makes a similar app for many major cities and it's great. GPS will work even without wifi. https://play.google.com/store/apps/details?id=com.triposo.droidguide.budapest
Szimpla Kert Ruin Pubs is a good example of a ruin pub
go on a hike on the Buda side
go to one of the many thermal baths. Many of them have specific days for genders s check before you go
Eat good food. Avoid going out to dinner with big groups, servers dont enjoy it
Wifi is hard to get leaving the airport so i used the airport shuttle a few times. 7-15 dollars/person depending on where you go but it will drop you off at the door and you don't have to worry about getting you rate jacked up by a cabby
I just spent the past month here as a US college student so hopefully i learned something about the city