I've done pretty much exactly the same thing, several times. I'm not sure I'd recommend that you go out in the countryside on the weekend (although there are some classic locations). London is one of the great walking cities on earth, and you could spend your entire photographic life there and not use it up. You could start around Trafalgar Square, walk up Charing Cross Road, make a detour over to Piccadilly Circus, walk around there for a while, go back to Charing Cross and walk up to the area of the British Museum (Bloomsbury), then take Oxford Street over to Hyde Park...Or jump around the interior of the city on the tube.
If you decide that you really need to get away to the counryside, instead of driving, which can be a serious pain and expensive, I'd recommend taking a train up to Oxford or something, which is fast, cheap and that way you can spend time looking at the countryside, rather than at the traffic. If you really want to get out in the countryside on the weekend, and you don't care which countryside, don't forget that Paris is only about 4-5 hours away by train...you can do a round trip in a day, if you start early...
But really, I'd stick to the interior areas of London. Great place. Your most important equipment will be a good pair of shoes...
JC