I have taken swim classes around here with my kids since they were about nine months old. I usually look for red cross certified instructors.
MOST classes will not teach start teaching kids to swim until about age three. Earlier than that, it's mommy and me swim classes. They teach you holds to get kids comfortable in the water and positioning for swim strokes. They usually teach water safety too...often a day of CPR reminders. You play water games and learn to blow bubbles. That sort of thing. I usually paid in the neighborhood of $65 for a two week class for mommy and me. Now that I am looking for classes for my three year old, it's going to be about $85 for 8 sessions. I have always used our local rec center. But the YMCA usually gets good reviews around here too. And there are places that deal exclusively in year round swimming lessons too (Aquakids is one here locally), but they are usually a little more expensive.
There are classes available that teach "infant aquatic survival" skills. Basically they teach babies to flip over and kick until they get to the edge of the pool. Apparently it is a hotly debated topic. I talked to my pedi about it. He said that it can't hurt, but those lessons are NOT swimming lessons, they are survival skills. He also said that many babies did not enjoy the survival type lessons and they could cause anxiety for some children. When I called and talked to a private instructor she said she definitely recommended the survival classes for parents with pools, but not to those who didn't have pools.
I have a six year old and a three year old. The six year old is an excellent swimmer for his age. My three year old can swim underwater like a little fish.