A great place to start the day if you are in South London is Federation Coffee in Brixton Village. The food is great and so is the coffee. This is a very chilled place where you can get both and indoor and outdoor experience (even though the whole place is under the Village roof).
The second coffee shop has to be Kaffeine. Having only recently come across this one, I was certainly impressed and they may even have the best coffee I've had, since a short stay in Portland, Oregon.
Nice old trees, lots of space, good for walking, running, hanging out with your kids and most of all, the perfect place to lie in the grass and chill out.