swimcat
I got my daughter in the water at 7 weeks. The city pools won’t have a problem with you bringing an infant under 6 months (babies under 1 are free too!), though I find the Seattle City pools are a bit cold but my daughter tolerates it. If you go to SW, a dip in the hot tub after being the pool would be a good idea to warm up.
For the first few swim outtings, I headed over to Bellevue’s lone city pool; there are two at their facility and one is a warm water therapy pool (kept at about 95 degrees). I like warmer water so I figured the kiddo would too! Bellevue charges for infants though, which I think is bull, but oh well. My baby didn’t cry and I was comfortable so it was worth the drive and expense IMO for about the first 6 or 7 swim outtings. Now we usually go to Medgar Evers since their family swim/lesson times are more convenient.
For diapers, I bought a reusable swim diaper. They are about $10; I bought one at Birth and Beyond. I use cloth diapers so I am not familiar with disposable swim diapers but I’m sure they are for sale at Target or drug stores.
It’s great you are planning ahead to get your baby in the water! Babies don’t do much the first few months in the pool, but they do get acclimated and comfortable in the water. At about 5 or 6 months they start kicking and splashing.
I don’t know of any special lessons for 0-6 months; the 6mo-3 year lessons offered by the city of seattle left much to be desired. It was basically a roped off family swim time with no instruction provided. I have an extensive swim background so I knew what to work on- if you have questions, just let me know and I can give you some ideas!