I also took my daughter to swimming lessons when she was 8 months old. I would actually suggest taking your stroller or infant car seat. I tried it both ways and it was a lot easier for me with the stroller or car seat.
Here's what you do when you are alone:
Wear your bathing suit under your clothes (BUT remember to bring underwear/bra for when you get out of the pool!)
Dress your baby in easily removeable clothing (no onesies/bodysuits)
Bring the stroller into the dressing room because the floor will likely be wet/muddy and without the stroller there is no where to set down your baby. The benches are usually very narrow without backs (locker-room style). Get youself ready first, then your baby. Sometimes the dressing areas will have a baby changing table on the wall in the handicapped shower area or in the handicapped bathroom stall, if so take advantage!
You can usually keep the stroller either in the dressing room or next to the pool.
As for entering the pool with the baby. This is the first lesson they will teach you! :) If for some strange reason they don't, just watch the other parents or ask the teacher. You basically hold the baby in one arm and hold the hand rail with the other hand. The lesson usually starts with an introduction outside of the pool.
When you get out, shower with your baby and take your baby's bathing suit off while showering. It comes off better when it's under running water for some reason.
Then change your baby into warm clothes plus a hat or hood to keep thier head warm from their wet hair. Then change youself and then you'll be on your way. :)
Have fun, I'm sure you'll love it!
P.S. The public pools in Washington don't allow disposable swim diapers. You might want to check your pool. I bought the iplay brand that is built into the swim suit.