I noticed your message cause I've also got a little girl who will be one in September, it would be great to meet up when you arrive! The organisation I've found most helpful is Playgroup SA, which supports playgroups and can tell you when and where they are. They've not been running during the summer holidays so I have just started going this week, one is held in a local kindy and its a nice way to start getting to grips with all this kids stuff!!!
I think its hard to recommend an area as it probably depends on what you're looking for. We live in Prospect which I really enjoy as there are lots of cafes and you can walk into town, I love chillin out under the trees with my baby girl in the botanic gardens!
See you when you get here!