At 125 or as a first big bike?
At 125, youll probably only need it for a few months so anything that will teach you the basics and costs under £500. Just remember, I've never met a biker that didnt drop or crash their first bike, so shiny RS125 plastics that cost £500 for each side fairing are probably not a great idea
Honda CG125's and Yamaha SR125's tend to be the norm, or offroad style bikes, like tatty KMX125/DT125's, all are seriously cheap.
As for a first big bike, the 500's are okay, but are generally got rid of within a year and traded for something with more oomph (costing you more in the long run), but if you have your heart set on one then the usual 500's are the Suzuki GS500, Honda CB500 (or CBF500 for a new one), ER-5 or a GPZ500. The 600's will probably last a few years and can cost from as little as £1000 for a 6-7 year old one that still runs well. Very cheap, but boring, is the Yamaha Diversion 600, the Bandit is a good bike, again fairly cheap, its not as fast as some of the newer bikes but still 0-60 in around 4 seconds and i embarassed a fat bloke on an R1 on mine, so miracles can happen! The Honda Hornet (CB600f, i think its called) has a detuned CBR600 engine, more power than the bandit but a really small fuel tank about 80-90 miles, very annoying. SV650 is a really nice looking bike, has a v-twin engine, different to the rest, low on insurance and handles really well, again fairly cheap (about the same price as a bandit).
I'd get a tatty little bike, pass your test and then get a 600, probably a bandit or a SV650. Both are good bikes that'll introduce you to biking and give you the thrill of some extreme performance without taking your head off. Remember to save about £500 for some good kit, at least a good helmet, leather jacket, good gloves and good boots, and preferably some armoured trousers, i've met a lot of bikers and none of them have managed to avoid crashing (including me, which is why my bandit is in pieces and i have more metal in me than is left on the bike) so make sure you protect yourself!