Of course you can drive any old vehicle you find on the side of the road. However, there are various ways in the game to unlock special vehicles; you can access them any time during the game from their corresponding shops:
Cars and other road and offroad vehicles can be found in the Garage.
Planes can be found in Hangars. Planes that can land on water can also be found in Boat Shops.
Watercrafts can be found in Boat Shops.
Helicopters can be found in Helicopter Shops.
In each shop you can scroll through the available vehicles. Vehicles that are greyed out have not yet been unlocked, which means you first need to meet a certain requirement in-game, for example complete a specific mission, to get access to the vehicle. The description will inform you about the requirement. Some vehicles that are unlocked need to be purchased with Dollars before you can use them.
If you have trouble locating content you should have unlocked, the information in this FAQ might help.