Well, if you're planning to study there, get the young person's travel card (or something like that) - can't remember exactly - there is the young person's one (for ppl under 26) and the student one. It does give you great discounts on admission fees to loads of places and some discount on travel I think.
Well, places to go in London:
1) since you're going with your mom, go to the usual Buckingham Palace, Tower of London, Tower bridge (can see Big Ben there as well and take pics), Marble Arch, Madame Tussauds etc.
2) If you're a soccer fan, visit all the footie grounds. There're loads in London. (You can do that in Manchester as well)
3) Go to Camden Market - it's Peninsular Plaza/Queensway shopping centre in a town format.
4) Portobello Road. If you're in London over the weekend, you'll catch the Portobello Market.
5) Go to Leichester Square - catch a play, go to Chinatown
6) Go to Bayswater - eat Lobster noodles at Royal China (may as well, since you mom's around) or eat the 烧腊 there. It just tastes better.
7) Go shopping - go down Oxford Street, go to Harvey Nichols, pop by Harrods.
If you manage to catch the summer sales, great discounts. The pound might be high, but you can get 50% off current season stuff and it still works out pretty good and it's made even better when you can collect back the VAT at the airport. Next (the Brit equivalent of Zara I guess) is also pretty good to go. Bring your mom to M&S, she'll have a field day there. You'll also discover that Top Shop/Top Man is very much like our Giordano/Hang Ten and we're really overpaying in SG.
8) Go to Hyde Park, spend at least a couple of hours there. It's lovely in the summer.
9) Go look for weekend flea markets, it's interesting to see what people sell there.
10) Take the London Eye - I'm not too keen about it, but if you want to get a panoramic view of London and have 15 quid to spare (thereabouts), go for it!
11) ***This is only if you like looking at Museums and natural history - The Natural History Museum, London. It's at South Kensington and it's FREE (mostly). Personally love it and think it's really good to go.
Manchester? Hmmm.... only been there briefly for a few days... all I did was walk around the Chinatown, the city centre, visit Old Trafford (ah... the folly of youth) - now prepare to be fleeced to get a view of anything inside the stadium, eat Indian food (loads there), so that's about all I can recommend.