Your title says "Drive" but you ask if you get taught to dive.
You won't be taught to drive in the Navy unless the billet you are in requires you to know how to drive. Then as far as I know you still have to go and learn how to drive but it may be paid for by the Navy but it may be different nowadays
Yes you can learn to dive in the Navy, through AT or become a diver
When I was in the workshops in Defiance (now Drake) there was a Leading Hands billet which required a full driving licence. A killick got drafted in who couldn't drive and when we tried to get him on a course he was promptly drafted out of the (cushy) billet and someone else sent in his place.
Just thought I'd mention it in case you thought the mob would readily pay out if you got a driving essential billet. It's best to bite the bullet and do your test before you join and come in with a full licence IMO