please no more fire fighting, save litten
I don't mind having fire/grass/water starters, but I think it would be interesting if they broke the tradition. Maybe with another trio of types that are super effective against each other, like rock/flying/fighting or ground/poison/grass. You'd still have a place to start for people new to the series learning the type chart, but also be doing something new for the people who have seen 'em all.
The three stage evolutions help to keep them balanced throughout the game, plus there's some nice symbolism there about growing up (keeping in mind that the canon games are directed at a younger audience), so that's something I would personally keep.
Since they're supposed to be unique, I wouldn't make them available in the wild. It is a bit of a big deal which one you pick, after all, especially if your rival(s) get the other starter(s). I would make it so that you can get the other starters through a post-game sidequest, though, for dex completion (or if you want to play around with them during the post-game).
I think they should stay with you throughout the entire game, but that's more of personal preference than anything else. Make them strong enough so that they don't punish sentimental players (like me) who would prefer to keep them, but I wouldn't leave the player screwed if they don't use one, either, for people who would prefer more freedom with building their team.