Take A Driving Course
Younger drivers can take the Pass Plus training course - it's a six hour course and several of the bigger insurance companies such as the AA, Endsleigh and Churchill do recognise it. You can attend the course even if you have points on your license! If you passed your practical driving test more than a year previous then do check with insurers that the Pass Plus is valid for a discount.
This is probably the single best way a young or inexperienced driver can get a sizeable discount on their car insurance. The course costs from around £100 to £150 yet can get you several hundred quid off your insurance price. Also check with your local council - a number of them offer discounts on participants of the Pass Plus scheme - win!
If you have taken the IAM (Institute of Advanced Motorists) test and passed, you can also get significant discounts from insurance companies (although not all do offer the discount). The course costs £149 so it could make very good financial sense to get this qualification under your belt.