Our son has been doing swimming lessons since he was about six months old (he is now two and a half) and for this entire time we have used these splash about baby wraps (in increasing sizes) to keep him warm and happy in the water.
They fit over a swim nappy (which is important to absorb any "emissions" which may occur in the pool) and are very easy to get on. The entire baby wrap folds out flat, you place your child on their back onto the wrap and then proceed to bring the flap through their legs, wrap each side of them and then stand them up and put the shoulder straps over. All the fastenings are securely held with velcro and the design leaves your little one's legs and arms free whilst keep their core torso warm using neoprene (like a wet suit).
I would whole heartedly recommend this product to anyone who is about to take their baby to their first swimming lesson...you'll stick with them too.
By the way, the manufacturer says that you should not machine wash them, but after a while we started chucking it in the washing machine with trunks, towels etc... on a normal wash at 40 degrees every week and (apart from the logo starting to fade) had no problems at all.