Does performing drive-by's increase my skill with the weapon used?

Solution 1:

Yes. I’ve already received notifications of my SMG skill upgrading while doing a drive-by.

You can verify that simply by checking your score with your current weapon before and after drive-bying a civilian. It is the first value in the “weapons” section of your stats.

Solution 2:

Grand Theft Auto San Andreas has a system called weapon skill. Weapon skill if a measure of, you guessed it, how skillful you are with a weapon.

Weapon skill is obtained by successfully hitting your target when you fire. The higher your weapon skill is the greater number of shots will hit their target.

Weapon skill is divided into three levels: None, Gangster (Obtained when skill is between 10 to 30% and Hitman (skill is 100%)

Your skill with a weapon can be seen from holding L1 when holding the weapon.

Some weapons such as melee and rifles do not have this weapon skill and this is mentioned under these weapons.

"To raise weapon skills fast, park a car in a garage, and shoot the centre of it's wheels repeatedly. The weapons that raise fastest with this method are the SMG, AK47, and Silenced Pistol. Although, all the shotguns seem to do very poorly with this method, and benefit much more from headshots on people."

The above is copied from this FAQ on GameFAQs.

Further sources relating to weapon skills in GTA:SA can be found in this thread and in this FAQ on gta-sanandreas.com.