What does it mean when you have to pick what regiment you go to?

It means that you have to choose a part of the army to go to - job / trade choice. If you have chosen your job / trade choice and you are asked to choose a Regt then it's basically location and what that Regt does (they are all ever so slightly different) - for example one Artillery regiment might be a close support AS90 Regiment and another might specialise in aerial 'stuff' (sorry not a Gunner but just trying to show the difference between what different 'Regiments' might mean). Hope that makes sense.

I would like to be a driver

Ok. You could be a driver in the Royal Logistic Corps or you could be a driver in the Royal Engineers etc. Have a look on the army website under role finder. You won't need to pick a Regt (ie place to go) until you're in Phase 2 training and you will be given options.

I want to be in the royal logistics