It would help knowing what neighborhood you are in, to see whether what I might suggest would be accessible for you.
I would start by checking out the Planetario Revista website, which lists a lot of kid friendly activities. I'm not sure what there is for really little ones but it's worth a look.
Also depending on your area maybe check into Gymboree or a local sports club to see if they offer any kind of kinder gym or movement class for toddlers?
If you pick up a copy of the Planetario Revista, in lots of kids' stores, you might be able to find something in your 'hood, a movement, music, or art class.
I notice at the playgrounds that Saturday seems to be the day when I see a lot more dads...what about posting in another forum to see if you might be able to get a Dads' Group going? I doubt you're the only dad with a toddler who could use something to do on a Saturday!
good luck