First you have to craft yourself a car key and a car. The car gets an identifier as soon as you craft it (you can check it by hovering your mouse over it). The key will show "not paired" status at start like here:

You need to place your car and right click with that clear key on it. Your key will get that car's identifier and they will be paired now. If you want to have more than just one key, you can craft a key modifier (recipe above) and place this key in first slot, a new (unpaired) key in the second and you will be able to use button "Copy" to copy key's identifier to the blank one. If you lose your car or just want to unpair a key to use it elsewhere, just put it in the first slot of key modifier's and press "Unpair" button.

When your key is paired with a car, you can right click when you're close to that car to open/close it. When car's lights are turned off it, of course, means that the car is closed. When you open it, you can get in and start driving. To start the engine, right click your key while sitting inside the car. When the engine is running, you're ready to drive! You can also use horn (by pressing H, you can change this key binding as well) for fun or for scarying animals (animals get scared of car horn about once every 20 seconds).

If the car is a truck, you can store items in it by pressing G while being inside the car (as passenger or driver).

Transporter usage tutorial:

Transporters are vehicles capable of carrying any mobs you can use lead on. To get them in your car, unlock it, craft yourself a whip and get some leads. Use the leads on animals and right click the transporter with a whip to make them enter it. To make a single animal leave the car, left click the car with a whip. To make all animals leave, right click the car with a whip while sneaking or just lock it with a key (as locking the car always removes all of it's passengers).