As I said earlier, you don't need a VAT number!
99.9% of private pilots haven't got one.
The only time you are likely to be asked for it is if you use a company name for the fuel etc. Quite a lot of private pilots own their plane in a company (I used to, years ago).
Re cost, you can work that out from the distance, the hourly rate charged by the school, and the speed. It's going to be a few hundred quid, obviously.