Hi! Right you know where Sainsburys is so you are already on to a winner. At the back of the car park there is Mere Green Community Centre. There are lots and lots of baby/toddler classes there. As well as Zumba/yoga. There is a playgroup on a Monday morning.
Have a look at Baby Senosry classes, they are a good way to met new mums as babies can go from very tiny. They are more for you than baby but nowt wrong with that!
Wyndley Leisure Centre has a baby pool if you want to go swimming as well as full leisure facilities. They used to do a buggy walk type thing - not sure if that is still on. It was just a walk around sutton park once a week I think. There is also a gymfants class there, as well as other kid's stuff.
I'm from Boldmere so know more about cafe's over this side of sutton, there are a couple of Boldmere shops and also Wylde Green shops. There is a Costa, again by Sainsbury's in Mere Green. Not sure about independants.
There is also a garden centre down the road in shenstone. They have a nice cafe and also soft play for when baby gets bigger.
There is a Bump, Babies and Beyond group in Emmanuel Church in Wylde Green - about 10 min drive, maybe 15, from Four Oaks. It is run by NCT and would be a great way to meet people with babies around the same age. It's a nice church hall too. warning netmums 
Any other questions please ask!