Eddy,
Your question is a very open one so it is difficult to know where to start. If you are in the UK then I assume you will be buying a motorcycle for when you are 17, passed the various stages of the test and can ride it. Then you have the power output to consider as new riders have restrictions.
Then you need to decide whether you want something from, say the 50s, 60s, 70s, 80s etc. or before/after. Do you want to buy it and ride it, buy it and get it going or buy it and restore it to perfection?
If you want to start an easy route then look for a BSA Bantam D1. They are still a relatively cheap 125, very simple to work on and there is quite a following. If you want Japanese, the Honda CB125, CG125 etc are a little more complex but won't break the bank. There is something for everyone but if you go for something obscure or rare for your first bike then you could be asking for trouble.
It really is a matter of your choice. Does something with loads of character but possibly a tendancy to break down appeal or something a little more bland but bullet-proof fit the bill.
If you are somewhere without the restrictions on young riders then there is a lot more scope but some countries restrict two-strokes.
Wherever you are decide how much you have to spend and buy the best thing you can afford, be it new-ish, old-ish or whatever. Then enjoy it for what it is. It might break down, you might have to fix it you'll probably fall off it a few times but your first bike is always special.
When I was sixteen I had mates that were in to mopeds then motorbikes the same as me. We learnt how to take bikes apart and get them running again (sometimes with 6 or so fewer nuts and bolts than they started with). We had near death experiences, broken bikes and bones and the best times I can remember.
Get something/anything and enjoy it! Best of luck.
