Most of the stuff I usually recommend to people has already been recommended – the Globe (rent a cushion if you’re going as a non-Groundling – the seats are hard!), the British Library, the Soane museum, Tate Modern, London Eye. Unlike many, I really disliked the Tower of London, as so much of it is Victorian reconstruction that it offends the historical purist in me. If you want London history, try the Museum of London instead, near St. Pauls.
I’ll also second the A-Z, Time Out and TravelCard recommendations. And whatever you do, don’t drive in London if you can at all help it. Take a bus, take a black cab (avoid the minicabs if you can), take one of those bicycle rickshaws, but don’t rent a car unless you’re planning on travelling outside of London. And even then, consider taking the train.
A boat ride down the Thames can be fun, as can the tour buses (the hop-on, hop-off ones offer the most flexibility) and the walking tours.
For my off-the-beaten-path tip, I’ll recommend the Chelsea Physic Garden, a tiny little botanical garden just off the river. There is an admission charge, and opening hours are EXTREMELY limited (last time I went they were only open Sunday afternoons), but if you’re here before the weather gets cold it might be worthwhile.
Places to eat: Muhib on Brick Lane for Indian food. There are also places up by Euston that do really good Indian cuisine, but I’ve forgotten the names. Paul on Bedford Street near Covent Garden Market for sinfully good omelettes. Ikkyu on Tottenham Court Road for Japanese food (a bit smoky, though). And a friend who visited recently went gaga for Pret A Manger’s sandwiches, which I also like but which can be pricy. And they make a crap cappuccino.
Outside London, both Bath and York are doable as day trips by train (<cough>when the trains run<cough>). (Speaking of which, casdave, how long does the coach trip to York take? It’s two hours by train!) Hampton Court is also an easy train ride from Waterloo and a great day out. Oxford is a cheap and easy 90-minute coach ride from just outside Victoria (but be aware the town closes down in th early evening, so leave London early).
That’s probably enough to keep you busy. Oh, and check out the Half-Price ticket place on the south side of Leicester Square.