The best beaches are the ones which are only visited by the surfers . . . I can name some : Prainha, Joatinga, Reserva, and Macumba . . . after that, you can eat on those restaurants on the beach (really cheap).
On Friday nights, go to Lapa, a really nice neighborhood that has a lot of bars, clubs, and live music. You´ll find really nice and cool people.
Saturday morning, you should go trekking on Perda Gávea. Ask for the cab driver to take you to Barrinha and then just follow the signs. Up there (about 1.5 hour trail), you´ll see a wonderful view of all Rio de Janeiro. Go down directly to Barra da Tijuca Beach!!!
Saturday night, you can go to Leblon (General St Martin St) -- really nice bars!!!
Quick Tips:
Try to make friends with any local!!! It would be easier!!! They are really nice and friendly people!!
Best Way To Get Around:
During the day, buses are a good choise, but don´t get them at night -- then, use cabs because they're safer!!! The cabs are not expensive, but you have to check the meter!! And pleaaase speak a little Spanish or Portuguese 'cause none of them speak English!!!