Is there any connections back to older GTA games? I don't think I've heard the names of the three main characters before.
In earlier games there were gangs, the characters in this game looks too old for that, but are they connected with the mafia? Or are they completely independent gangsters?
There are only very slight connections to older games. I'll say now that there are very mild spoilers in this post, but it doesn't give anything away about the story itself, just minor details of a few missions. Having only completed 50% of the story, there may be other connections and details that I don't know about.
Connections that I've noticed are (obviously they're very small ones):
The game is set in San Andreas, but it's changed a lot since the game San Andreas was out.
There is one particular mission where you go back to Grove Street, and they mention the Grove Street Families gang.
There is a very vague mention to Niko from GTA IV, and Packie McReary from IV also has a very small cameo.
One of the gangs from San Andreas is still around.
As for the characters, one of them is a former gang member who wants to move up in the criminal underworld (but still helps out his old gang from time to time). The other two, as far as I'm aware, have no connection to the mafia or a gang. They're just old friends who used to commit crimes together.