No idea what it's like as my DC were older when we moved here, but I live near the Little Starz Childrens Centre and that runs new Mum groups. They seem really friendly - they do fireworks every November that's open to the local community and we've popped in a few times
My youngest has just turned 2; we really like the playgroup in the church half way up Gypsy Hill (can't remember the name) - it's on a Thursday morning from 10am + is best to be there on time. We've also tried the Salvation Army one, and the Chatsworth Baptist Church. All of these are really friendly and offer free play, crafty stuff, singing and a snack time.
We also really like the Peter Pan playgroup that's in the Peonix centre next to Sainsburys in Crystal Palace - it's Tues + Fri 9:30-11:30 and is really relaxed with loads of different things for little ones to do.
The children's centres all offer great groups on different days - try the Cherry Tree one on Barston Road, and Hitherfield School on Leigham Vale. I also went to a free paediatric first aid course at Hitherfield, which was great :D