you're best bet for a lot of spots in a small area is the city (1 square mile where a lot of business/finance stuff is located) some spots you'll get booted from, some you won't, you just have to chance it.
Just for the sake of clarity (for the op's sake), in London, though it's not really a big deal, 'the City' and 'London' are differentiated. Basically what (I think) he means is in and around Aldagate (the business district) there is an abundance of skate spots.
london is good for skating but it's more about lots of little spots as opposed to big main spots where's there's people skating all day (with the exception of southbank but that's more like a zoo/tourist attraction than a spot these days)
True about Southbank. If you're having a bad day it's not the ideal place to be as there are hundreds of people taking pictures of you, staring at you and just generally regarding you as some new installation at South Bank; take it as a compliment or an insult, your choice.
London has great museums, I would recommend the V&A museum in South Kensington.
Also with regard to skate stores, Slam City Skates is pretty central. Go to Covent Garden and get to Seven Dials (it's pretty well known) and from there you should be able to find Neal's yard quite easily.
Enjoy, London is great and has so much to do.