Oyster Cards with Pay As You Go (PAYG) credit are not valid on South West Trains services (i.e. services to Twickenham).
A map of National Rail routes they are valid on can be found at:
http://www.nationalrail.co.uk/times_fares/london/OysterPAYG.pdf
(Note the map shows acceptance of Victoria - Balham, slightly prematurely as issues are still being ironed out there and Oyster PAYG is not yet valid on that route).
If you are planning to travel on routes not covered then a travelcard may be a better option. A weekly travelcard season can be put onto an Oyster card (if purchased at a London Underground ticket office) or paper stock (if purchased at a National Rail ticket office). This allows unlimited travel on Tubes, trains, and DLR in the zones valid, the entire Croydon Tramlink (if the travelcard has at least one of zones 3, 4, 5, or 6 in its validity), and all London bus services (irrespective of zones held). This will also entitle you to 1/3 off most riverboat services.