This largely traffic free bike ride runs along National Cycle Network Route 21 from Beckenham to the Cutty Sark on the Thames. It's about a 7 mile run with waterside sections along the Pool River and Ravensbourne River. As it takes place on mostly traffic free paths, walkers can enjoy most of the route too. It's a nice easy route, suitable for beginners or families.
The route starts in Cator Park, Beckenham. Then head north through the park to New Beckenham, Bellingham and Catford. You then head through Lewisham and Deptford, finishing on the Thames by the famous ship.
To extend your exercise you can head to Greenwich Park just to the east of the Cutty Sark. There's nice cycling and walking trails to be found here. Around here you can pick up Quietway 1 which takes you all the way to Waterloo.
You could also continue north along NCN 1 and visit the Isle of Dogs on the other side of the river.
Heading south along NCN 21 from Cator Park will take you to South Norwood Country Park.