Dulwich feels like a mistake of geography. A proper village in Zone 3, with an eighteenth-century picture gallery, a decent pub, and actual green cricket grounds — all reachable by Overground in fifteen minutes from central. Four kilometres, gentle, mostly flat.
Start at West Dulwich. Ten minutes through residential streets brings you to the Picture Gallery — Britain's oldest purpose-built public gallery, designed by John Soane in 1817. It's small, sharp, and the Poussins are worth the trip on their own.
Across the road is Dulwich Park. Loop it clockwise; the American garden in late spring is genuinely unexpected in south London, and the boating lake has a tea hut.
Out of the park into the village proper. The high street is short and entirely credible: bookshop, bakery, butcher, the Crown & Greyhound (which is vast and fine). Walk past Dulwich College — a Victorian brick-and-terracotta pile that looks like it was rendered into CGI — and then north to the station. The whole thing could be a Sunday in a market town.