16yr old = learner bike up to 50cc
17yr plus = any learner bike up to 125cc
This was the rule in the 80's. My first bike was a 90cc and I waited until my 17th birthday for it. As far as I know this hasn't changed. Today you can learn on a 125cc if over 17. The exception is if you're on a Direct Access course where you learn on a 500cc once you have passed the CBT on a 125cc.
The official information is here, it's more or less as nigeltm says but as the government is involved it's a bit more complicated:
http://www.direct.gov.uk/en/Motoring/LearnerAndNewDrivers/RidingMotorcyclesAndMopeds/DG_4022568
http://www.direct.gov.uk/en/Motoring/DriverLicensing/WhatCanYouDriveAndYourObligations/DG_4022547
Asa, most of my family have been bikers. Back in the 1960s, when I and my cousins learned to ride, it didn't matter what bike it was, we just got on (and fell off) whatever we could get hold of. It doesn't need to be flash or expensive, it just has to meet the regulations and go. I'm sure whatever you get you'll love it.
I've done around half a million miles on motorcycles, I've toured most of Europe and I often travel from England to the Mediterranean coast on a bike. I have an advanced riding qualification and I currently own four bikes. Riding a bike is a real pleasure, the freedom it gives you is incomparable.
But please always remember that riding a motorcycle is the most dangerous thing that you can do in peacetime. A single mistake can all too easily leave you in a wheelchair for the rest of your life. When we were learning, my youngest cousin put himself in hospital so many times we all said we were going to buy season tickets. He broke his back once but he got away with it. It might not even be you that makes the mistake. Just over ten years ago, while I was on my Hayabusa, waiting at a red traffic light on the A38, the guy behind me ran over my bike with a Transit van. Fortunately I noticed early on in the proceedings and jumped off the bike before he managed to get me. A couple of people have tried to knock me off a bike deliberately, for no reason I could fathom, and a van driver in France threw his door open into my path (he thought it would take me off the bike, but the 250kg bike took off his door). When I was 19 a little old lady in Brighton threatened me with her brolly because I was wearing leathers. She said they didn't want horrible people like me in her town.
Take your time. When you're on a bike, think of yourself as invisible. Keep your wits about you, and make sure that you're always in control of the situation -- never the other way around. Go on as many courses as you can, there are loads of free ones that are well worth spending your time on. If you can, try to get on courses run by the police. I've ridden with many, and there are no better riders anywhere in the world than in our police force.