Oyster Card will save your life - it's brilliant

As a little bit of traveling advice, most Londoners moan about the Tube. That's because they all try and use it at the same time ( 0830-0930 and 1630-1800 ) - just avoid using it at those times and it's fantastic
Greenwich is nice - sadly the Cutty Sark isn't visible - but the Naval College and the Observatory are ( I was there only a couple of weeks ago ). If you take the tube to Canary Wharf, there's a nice short walk to the DLR line, so you can have a walk around Canary Wharf itself.
I would use the Tate-2-Tate ferry ( from the Tate Modern to the Tate Britain in Pimlico ) - that'd be a nice day. Also - the British Library (near Kings Cross and Euston ) is a nice hour. The Science Museum, National History Museum and Victoria & Albert Museum are all within 100 metres of each other, all free, and all great. The British Museum is a tube ride away, but is amazing as well and, again, free.
A nice walk is to get off the Tube at Waterloo - and then walk along the embankment, past the Eye, over the bridge, past the Houses of Parliament, and up to Trafalger Square - then onward travel at Charing Cross.
As for Cirencester - I used to go to school over there. The train journey ( from Paddington isn't too bad ) BUT - you have to get off at Kemble, which is in the middle of nowhere, and it's about 3, 4 miles from Cirencester itself. And if I'm honest, there's not a lot IN Cirencester. If you can get Car Rental somewhere west of London, and do a drive, get off the M4 at Swindon and go to Cirencester, Tetbury and then down to Chipping Sodbury. On the way out of Tetbury, you'll go past Highgrove house, and Westonburt Arboretum and then at Chipping Sodbury, back onto the M4 and East back to London. If you wanted extra-England - then at Cirencester carry on and head towards Stround and go thru Rodborough Common, Minchinhampton, Nailsworth and Avening. If you REALLY want to see some English countryside - have a gliding experience at the Cotswold Gliding Club at Aston Down outside Minchinhampton
Doug