Greenwich Park is the obvious starting point, obviously. But here’s the insider tip: skip the crowds by the Royal Observatory and walk down to the Flower Garden. It’s quieter, and you can pretend you’re in a Jane Austen novel.
For a wilder vibe, Blackheath is basically a giant, open field where dogs run free and people fly kites that look like distressed eagles. Bring a blanket and a smug sense of superiority. (You’re welcome.)