On the above, Arbutus and St John Bread and Wine seconded: both excellent. Wouldn't bother with Criterion: a famous old restaurant and wonderful space that has been fairly well buggerised.
On other food/drink related options, Borough Market is wonderful though perhaps best avoided on a Saturday afternoon as the crowds are getting farcical: they were dropping tourist coach parties off on my last visit and it's not a big space. Few great boozers (sorry, pubs) around there too: the Market Porter and the Globe in or adjacent to the market itself. And there's The George, on the other side of the Borough High Street and down a bit, which is surely London's most spectacular pub?
Liberty's meanswear dept was a bit smaller than usual due to building work on a recent visit. If you're in Piccadilly, Fortnum and Mason's is worth a schufti.
Try starting at Piccadilly Circus tube, walking down Piccadilly, via Fortnums, to take in Jermyn Street, up Burlington Gdns, into Bond St and Savile Row, onto Selfridges and then, bracing yourself, up Oxford Street to Liberty, then down Regent Street or, if you're all chainstored out, down one of the Soho backstreets like Wardour Street back down towards Piccadilly Circus again. If it's evening when you get done, continue onto Trafalgar Square for a a drink in Albanach (if you like whisky). If you're in a wine mood, try Gordons Wine Bar in Villiers Street (to the left of Charing Cross station), or the Ship and Shovel if you want a beer (an alley of Villiers Street).
British Museum's my favourite, for the building as much as contents. I also enjoy the V&A, the Tates (Modern and Britain, get the boat between them) and the Natural History Museum. Houses of Parliament are worth look though check in advance as access is restricted - you can't just rock up and pay on the door.
On the aimless mooching side of things, the South Bank is probably the best (Westminster to London Bridge - great views and, if you end at London Bridge, there's nice places for the pint you've earned). Also, the parks (St James and Hyde), Soho, Bloomsbury and Fitzrovia (don't miss Lamb's Conduit Street), and Kew Gardens, Greenwich, or Limehouse if you can be arsed travelling out of the centre a bit.
Sorry for droning on: just moved back after six years in Australia and I am enjoying being back here.