The Burgess Pool in Menlo Park is a group of heated outdoor pools with 1 25 meter pool, a 25 yard pool, a children's wading pool and locker rooms.
The Burgess Pool is great for little kids who are just learning how to swim as well as for the more advanced swimmers. There are 3 pools to splash around in. Last summer, my son was only 1 and we spent most of our time in the kiddie pool walking around and sitting on the steps. He'd usually work up the courage to go to the big kids pool and sit on the side and jump into the water aka my arms.
The pool is really kid-friendly and they allow for kid floaties, so that I don't have to hold him the entire time. I would bring one of those inflatable rings and plopped him in that and give my arms a break.
You can't eat near the pool, but there's plenty of picnicky grass areas in and around the pool since it's located in Burgess Park. Locker rooms are clean, although it's still a locker room.