I've been an expat in the US - was there from 1999 - 2003, and moved out there with four kids when my eldest was 8 and youngest was 6 months; we came back with five!
Mother/toddler groups are not quite the same as here, I'm afraid. You might be lucky to find a drop in place, but most mother & toddler activities are highly organised, where you pay for a whole term at the start. BTW, a place where you take your child and stay with them is called a Playgroup - but this could be anything from a coffee morning in someone's house to a big thing in the local community center.
I would suggest that you go to something like the local library's storytime and ask the moms there about what else there is in the local community. The other place is to join a church (but American churches are very differnt to the ones here).
There are lots of places where you can leave your child in activities or the nursery (ie creche), while you do an organised adult activity or your own thing.
Depending on the ages of your kids, you may be able to get involved in recreational sports. If they are, say, 4+, they can start out on soccer and baseball. This is a great way to meet other people and they are usually looking for volunteer coaches, assistants, and team moms. This kind of stuff would be run by your local city, or by the YMCA.
For the summer, it is really important to join a pool club. Your kids do swimming lessons and you generally hang out. You may need to put your name down well ahead of the summer season.
Good luck!