To start a date with a girl, simply go to her house.
Some girlfriends will like you to drive fast or slow. The girls have a lot of health but they can still be killed; if a girl is killed by CJ (not counting other drivers etc.
Each type of date has specific objectives, and each girl's requirements for the date is different.
As in life, your girlfriend will want to decide what to do in the date. In this date, which normally comes up first, your girlfriend will want to go out to eat.
San Andreas introduces the ability for the player to pick up dates and go out with them.
The main reason to date a girl is to get a special bonus in exchange.