DD, Little Miss CFH has used mini-westuits since she was about four months, like the wrap things you suggest (Peter Jones/John Lewis, natch) and they've been brilliant in the outdoor pool this summer. In fact, they've been utterly marvellous, as she loves being in the pool and it keeps her in there longer!
Indoor we tend not to use them, as the pool at our gym's kept nice and warm during kiddie friendly hours.
Floaty things? Well, we bought her a sit-in ring thing to go on holiday, which she adored, as she could splash around to her heart's content. It's been used a fair bit in the pools as well, so may be worth a look. Other than that, we do a lot of holding/supporting and encouraging her to kick, and splash away in the pool. Also got her used to being underwater very early, and she now has no fear! Dives in from the edge, too! We didn't bother with lessons, as both are strong swimmers, and we have a friend who used to be a kiddie swimming coach give us lots of tips.
So, neoprene suits are a yes! Floaty things are a maybe. Lessons are up to you.
It's great fun, though, isn't it! (Oh, and added YummyMummySwimmySpotting potential for the father!)