London has its good points but I think its a tough city to live in as a young person. There are a lot of things to do but you don't feel like you can make the most of them.
Househunting and accommodation issues are always a big bugbear. If you have to find a new place to live there's always a lot of hassle, going round being 'interviewed' by potential housemates that have loads of people coming round viewing the room so are in the position to be choosy. Then your rent eats up a huge proportion of your wage but you look around at the student-style accommodation you live in and think WTF am I paying for here.
Then there's the time factor, you are likely to spend a bigger proportion of your time commuting than in other cities which lengthens your working day and contributes to the general feeling of having little free time to yourself. Because London is so expensive, you need to be earning a pretty high salary to feel like you can afford to make the most of it but the people in London that do earn high salaries generally work very long hours so the more the financial pressure eases, the more your free time disappears.
There's an 'exclusivity factor' about a lot of the social scene that isn't there in other cities. When I was a student in Leeds, I felt that on the clubbing/going out scene everyone in the city was equal, you could sample all the places and naturally gravitate to the ones you knew best. But in London I feel like a lot of the clubs/bars are really for a clique of people that are 'somebody' famous - if you aren't at least on the fringes of the entertainment/sport/modelling industries you're nobody. On the flip side of that there are loads of pubs and bars so you can go somewhere new all the time rather than being stuck in the same old place.
Continuing with the going out theme - two major bugbears of mine. One - if you go for a drink in a pub after work the odds are you will be standing, often outside. Unless you reserve an area before you won't get a seat. Big difference from other cities when you can spontaneously decide to go for a pint with your mates and actually feel you are 'winding down' by being able to sit down and chill out and have some space. Two - if you have a night out, the logistics of getting back require major planning and you will lose people at 11pm because they want to get back home on the tube etc. Also if you get a taxi it will often be as expensive as the rest of the night out.
There are good points in London and the good things are that career wise if you are in a decent job with a decent field, you have lots of prospects in London and opportunities to move around and develop your career. When firms have multiple offices its a good bet that the London office is where its at and this is where you get a name for yourself.
But all in all I think as a young person if you can get a decent graduate job in a fairly large city outside London (particularly somewhere like Manchester or Leeds) then you can enjoy a higher quality of life and have better prospects for settling down and getting property etc. I have friends earning say £30k to £35k in those cities and they live great lifestyles with lots of disposable income to do stuff and then travel in to London to visit friends anyway.