It's a double-edged sword for me. Part of me likes a younger Peter calling people things like "Mr. Stark", "Mr. Castle", etc. It feels totally appropriate for someone of his age. I wouldn't even mind a little of that in this next movie, but there does need to be a clear progression. I think that was probably just part of his upbringing with May, and is probably a little hard to shake, but as he realizes more and more that he's essentially on his own, I'd like to see him start to shed that and move into a more mature (but still wise-cracking) version of himself. If anything, I think Punisher is an interesting person to pair him up with. Let him kind of explore that darker, more mature side, maybe go a little too far at one point, and do the thing that Frank seems unable to do, which is back off a bit and learn to balance that rage.
Depending on how they play it, I think Peter probably would want to join a team of some sort. It's the first time in a while that not only is he on his own, but nobody knows he's Spidey. There's upsides and downsides to that. I think he'll probably like the anonymity a bit, and maybe want to form relationships with other heroes just so he'll have a friend, but I do think the point of this new arc- whether it's confined to this movie, or it becomes a trilogy, whatever- is that we'll start to see a Spidey that's more independent, and works well as part of a team, but isn't necessarily the "intern" of the team, so to speak- he's more of a leader or team player.