If you passed your driving test after 2001, you'll need to go through Compulsory Basic Training to ride a 50cc scooter on the road. You also need to display 'L' plates, even after you've passed the CBT test.
If you passed your driving test before 2001, you don't need to do the CBT, you don't need to have 'L' plates and you can even carry a passenger. You must take the CBT if you want to ride anything larger than a 50cc scooter.
To ride your scooter legally on the road, you'll need to register it with the DVLA first. Normally, you'll get your scooter's registration documents when you buy it. If you need help, the DVLA have a dedicated phone number for questions about vehicle registration.
Just like a car, you'll need to tax your scooter to ride it legally in the UK. Small bikes, like scooters and mopeds, are often under 150cc and so they fall into the lowest tax bracket. This can cost as little as £17 per year.
Lastly, before you flip your visor down and hit the road, make sure you have insurance for your scooter. LV= offers comprehensive cover in the event of an accident or if your scooter is stolen. Your helmet, leathers, sidecar and accessories are covered too, as well as an optional £100,000 motor legal expenses cover, including help with personal injury claims.
You can find more information about your scooter licence on the UK Government website.