I used to live in Testacccio, so I am partial, but it is considered a "true" working class Roman neighborhood (though not so much now with reale estate prices the way they are). It has the historic AS Roma fan club, a great open market, and some great restaurants including my favorite pizzeria in all of Rome, Il Nuovo Mondo in Via Amerigo Vespucci. Yes there are the night clubs, all located together in one part of the area called Monte Testaccio, definately worth checking out, one after another,though things don't get going until after midnight. While you wait you can go to one of the nearby cafes or restaurants. Just know that in the summer many of the clubs move out to the beaches in Ostia and Fiumicino to avoid the heat.
During the day the Protestant Cemetary behind the pyramid is so neat, with tombs of people from all over the world, including many famous artists, intellectuals, and poets, but the one thing that they all have in common is that they died in Rome. My favorite is the tomb of one woman where it is written: Born in Stubenville, Ohio Died in Rome.
It is also worth walking up the Aventine hill which is directly over Testaccio and is one of the most beautiful neighborhoods in Rome with villas and churches, and a fantastic view of the city from the Orange Garden next to Santa Sabina.
As for Rome in general, there is something going on everywhere at night, but check out Trastevere and the San Lorenzo area, those areas have lots of night life.