Dogenzaka said:
Stay away from the food. It is absolutely disgusting.
That's so not true. Tons of amazing restaurants in London. Plus if you're a vegetarian it's like heaven. London has been named the most vegetarian-friendly city in the world and I agree (of course you've gotta know where to go!).
Even if you aren't vegetarian, there are tons of good restaurants, just look them up online on Yelp or whatever.
I haven't seen anyone talk about Old Street/Shoreditch. It's hipster central and if you're not put off by it, there are tons of really good places there to go out at night. I guess it's kinda like Williamsburg in NY, very Brooklyn-ish anyway. There's a place for brunch in the morning called the Breakfast Club that is kinda cool. If you wanna chill it's definitely a good place (there's one in Angel and Soho as well but I've not been to any of those). Lots of people though, so if you go in the weekend be ready to queue a bit or arrive before 11am.
If you can, make sure to get a proper English tea, with scones, finger sandwiches and shit. They are usually expensive but it's worth it.
Fabric and Ministry of Sound are shit. No offense to people who like it but I've had terrible evenings in both of those places.
I dunno it Shunt is opened or not these days but it might be worth checking it out. I also like Koko but it's a pretty young crowd and I felt old last time I went (and I'm 24 :3).
If it's sunny, you should check out Richmond. It's far in zone 5 (6?) but it's really pretty and you can walk along the river, shop and why not even swing by Kew Gardens like someone suggested.
Greenwich is a good idea too, especially since you'll have a nice view of Canary Wharf and the London skyline up on that hill.
There's also a roller disco night at the Renaissance Rooms in Vauxhall, supposed to be fun so why not try that if you're into roller skating and shaking your booty.
I'm guessing you're straight so I won't list you some of the cool gay places I know but Soho can be fun at night even without being gay.
There also is a all you can eat sushi place there if you want. Food is not that good but if you wanna get filled up with ok sushis it's called Hi Sushi.
What else... well since you like theatre and shit, make sure to go see a play. I won't advise you because I don't know shit about it but it's worth it. Musicals are legion too, but I guess you have Broadway back home so maybe that's not interesting for you.
When you are near Picadilly Circus, be sure to go down Picadilly (the street), there is a department store called Fortnum and Mason that's worth seeing. Plus they sell good (expensive) tea if you want to bring some back. Buckingham Palace is in the vicinity as well, I think it's worth seeing at least once.
I haven't been out of London that much (which is a shame since I've been in the UK for 2years and a half now) but I remember Bath and York as being cute and worth a day trip.
As for Paris, the train makes it really easy to go for a couple of days. You take the train in the centre of London and get to the centre or Paris so you won't have to transit from the airport. Plus if you do it during the week you should be able to get return tickets for £50. Might be a good idea!