I took Joey to Mommy and Me swimming when he was a baby. The first time, he screamed bloody murder. Don't be deterred!!
The next time, he was just quietly miserable, and the third time he did great! I do agree that the warmer the pool, the better. Basically, the shock is that Mommy is submerging them in something frighteningly unlike anything--and yikes! *I* don't even like cold pools.
I'm not expert, but I personally would hold off on the water wings. I just held Joey the whole first time, and slowly got him used to swimming. Together we worked on kicking and paddling (the instructor helpfully called it "scooping" the water). Then, and this was a biggy, we had to help them get used to lying on their back, with their head resting on Mommy's shoulder. This is very hard for them, it's like a little trust game or something.
The most important thing we did was teach them where the wall is, and how to grab it. Joey was little, and at first I felt like rolling my eyes, but he DID remember each time we went back. Not that I expect him to do a butterfly over to the wall in a moment of danger, but if it's possible, it's definitely a plus if they can grab on and wait for help.
Also, sitting them on the side and then "jumping" in. The surprise of going from out of the water to in it is a big deal.
And then at the end, it was a huge deal to just let them play and have fun. Seeing if they reach for a toy in the water, getting splashed by a squirty toy, etc.. And of course, SPLASHING!!! It's probably why Joey tolerated the class!
Hope this was helpful.