And Jet13, if you are going through all the trouble to learn a little Hungarian, i will assume that you really want to get more than tourism out of this. So:
Avoid Vaci utca
Go to an Operett or a Opera. Get excellent seats and dress for it. At the Operett the center boxes are called VIP boxes and you get taken to the parlor at intermission for drinks.
Go to a Jazz Club. I like Opus.
Avoid tourist "Traditional Dance" shows
Avoid tourist Gypsy Concerts
DO go to a restaurant that has a Gypsy band.
Visit the upper half of District VIII
Saturday morning go to service at a synagogue
Avoid Ruin Pubs late at night (unless you like drunk Brits)
Go to Kadarka and tell them you want to learn about Hungarian wine. They will set up flights for you.
Go to the Great Market Hall, but also go to one of the still "real" market halls. Be sure to check out the basement at the Great Market Hall
Do the short tour at the Great Synagogue and be sure to visit the Rumbach synagogue. (look for a side door)
Buy something from the Jewish bakery on the same street.
Friday night go to the Klezmer concert at Spinoza's, unless then Budapest Klezmer Band is playing someplace in town, in which case do what ever you have to do to be there.
Don't go to the Shoes on the Danube memorial to gawk.
DO stop at one of the hundreds of sidewalk florist shops and purchase a small simple flower, take it to the Shoes on the Danube Memorial and place it with a shoe.
Take a cooking class in the home of a local.
Walk into as many open court yard doors as humanly possible.
Stop at every Stolperstein and read the name out loud.
Go to a bath house. Any bath house.
Late at night, the night before you leave Budapest, get a bottle of wine and a couple of glasses and take a taxi to the top of Gellert Hill, on the left, just before you reach the Citadella there is a park from which the famous view is possible. The taxi driver will know where to take you. Wait time for a taxi is 70 forints a minute so a 30 minute stop will cost you about $8. Throw in the round trip fare from Pest and you might spend $30 on the event. Worth every penny.
Visit the cemetery: Kerepesi Cemetery (Kerepesi Temető) and if time permits try and get into the Jewish cemetery behind it on Salgótarjáni utca
Do eat gulyás, Töltött Káposzta and Somlói Galuska; but do it in a place that terrified you to walk into.
Do ask everyone that speaks to you their name and thank them by name.
Do look around to determine where in town you want to stay next year.... (prediction).