If you passed your car test before 1 Feb 2001 you can ride a 50cc bvike straightaway. For anything between 50-125cc (or later car test) you need to take and pass the CBT - that's one day training and test and pretty straightforward. With it you can ride but you need L plates and can't carry passengers or ride on Motorways etc. Within three years of that you'll need to take a full test. Well worth it!
As for the scooter of your dreams - the 'real thing' would be a Lambretta or Vespa - the latter still produce the PX125, which is *the* classic. These bikes have gears. Or you might prefer the imitations - Havana etc, which usually have automatic gearing. Me? I ride a PX125 and, even though I'm the slowest bike on the road, I feel the proudest and undoubtedly look the coolest!!