I actually think it would be more interesting if Snape failed to catch Wormtail and he was the one to wind up in Azkaban. Sure, Sirius knows that Peter was the Secret Keeper and all that, but would he reveal that?
After all, it's his old arch-nemesis Snape who gets to suffer and being a former Death Eater and all, he could probably expect even less mercy then Sirius if he were to be framed for those murders, magic in public and whatnot.
Lots of moral dilemmas in such a twist, which is something I like to see in a story.
Anyways, back to the original idea.
Well, obviously Harry wouldn't be Harry anymore. With such a radical change in his upbringing, he'd be a *vastly* different person. I can picture him having a great fondness for pranks, or that he got so fed up with 'em growing up that he's now radically opposed to them ( albeit probably extremly good at spotting them and working out the mechanics behind them ).
He wouldn't be ignorant of wizarding society and customs, as I hardly think Sirius would go all-out muggle. On the other hand, he'd probably be ignorant of muggle society to a much greater extent then canon Harry, though probably not as much as most Purebloods as Sirius seems at least somewhat aquainted with the muggle world.
Hogwarts probably wouldn't be the first time he ever sat on a broom, as I can imagine Sirius encouraging and nurturing even the faintest hint of interest in Q.
He'd probably know at least a bit of magic, perhaps not more then a handful of spells, but I don't think Sirius would have waited until it's time to start Hogwarts to get Harry a wand.
But the changes to focus on, would be Harry's personality and behavior, which greatly depends on how well Sirius handles being a parent. I don't think he'd make a very good one, to be honest and chances are that Harry would be a spoiled brat somewhat along the lines of Malfoy.
On the other hand, if he's not cooped up in prison, mayhap he's managed to land himself a wife or a 'close male friend' who can balance out his corrupting influence on a growing young boy.
Remus could be a calming influence, pending on how accepted werewolves are. If it's just general dislike of them, nae problem. If it's more then that, it'd probably attract too much attention if a werewolf spent too much time around the boy-who-lived, so his influence would probably be minor at best.
Anyways, first port of call when writing this fic, should be to determine just how Harry is different from canon and then go from there, working out a story from that.
At least for character-centric ol' me.