Can't help with Manchester airport but used to work at Stansted.
Misskits is right that normally to work at an airport you need to be at least 18, occasionally we took people younger but not very often, wasn't involved in the recruitment so don't know the whys and wherefores.
You might struggle if you are relying on public transport, as airport check-ins are 24 hours, and if its anything like stansted the normal starting time for check-in was between 3am - 4am, and likewise there is normally late finishes too. Just something to bear in mind!
My training, I think (this was about 8 years ago now) was about 2 days in a classroom, then a week or so shadowing to get up to speed, however I wasn't working on check-in, I was out at the departure gate, and later became a dispatcher (person who turns around the aircraft), for which further training was needed - think that was 2 weeks, it was quite complicated! I would imagine it would be something similar for check-in, a couple of days in a formal environment, followed by a period of shadowing.
Shift work is hard, and not everyone is suited to it, however, in the most part I enjoyed my time at Stansted, was certainly an experience! Gives you a chance to explore your skills, which you can then use in a variety of other jobs.
Good luck with your jobsearch. Post again if there is anything else you'd like to know and I'll try to help!
kind regards Polotoo