The Heart Foundation Shop on Old Kent Rd. They have new stuff in every week - sofas, tables, chairs, wardrobes etc - it's really good if you're just starting out. I furnished my first place from charity shops and it's usually pretty good quality (& v.cheap of course) plus you get the added benefit of feeling self-satisfied that you have helped out. They say that they will pick up stuff, so I'm sure they'd probably deliver (maybe a small fee?). Good luck!
There is no such thing as an Athiest on a Battlefied ... Anon
Yeah, seconded vis a vis the Heart Foundation place on OK Rd. The quality can be variable, and often it's just full of 80s MFI stuff, but I have managed to find quite a few 50s/60s/70s design pieces there for a few quid. It's worth popping in here on a regular basis if you can, as things fly out of this place very fast, but they can also have quite a quick turnaround of stuff.
Bermondsey Market is a lot more expensive than charity places, but also worth a try (plus the shops in that locale) although you have to get there very early, which can be a bit of a test...
Also, surprisingly, EBay can be quite a good place for buying the odd small/design-conscious thing, if that's your bag. Often very low prices.
There is a place on Grange Road (not far from Tower Bridge Road). Never been in, just see it as I am travelling down the road. Seems to be of the house clearance sort - with stuff piled outside on the pavement - hopefully the sort of place where undiscovered designer pieces can go for next to nothing. Seems to be open 9-5, mon to fri but cant be any more specific