Budapest is a cheap city so having a budget night out is not hard at all.
One can start from the many restaurants in the east of the city, they are small, family run and skomey. The atmosphere is friendly, the waiters are smiley an the food is warm and filling, don't forget to try some hungarian goulash and some hungarian red wine, it does not have a french label but its an up and coming type. The centre is full of bars and clubs, to get in it is about 5 dollars. Try Flasdance in vati cuch to get to know a local place or alternativel the Fontana venue for some up and coming hungarian band all for less than 10 dollars.