I'm not quite clear on what you are trying to achieve. Are you just wanting more lighting when you pull your vehicle into the garage and walk inside or are you wanting task lighting?
I'm looking for more light and in additional locations that will come on when the doors are opened by a car.
The garage has ~10 foot ceilings and the garage door openers have lights on one side. To have matters worse there is a beam near the openers that blocks a fair about of light. Even with brighter LED bulbs in the housings the areas of the garage that are dimly lit are where you have to walk to get in and include stairs and seems to be where people like to leave things lying around (or missed when throwing something into the recycling).
My garage door opener comes with a light switch on the wall mounted open/close switch so you can turn the light on and it will stay on until you press the button again or open/close the door. If you don't have that feature available, perhaps you can upgrade.
I do not have that feature on my openers, but I do have an existing switched lighting circuit consisting of 5 utility light fixtures and LED bulbs which provide more light than the garage door openers. Presuming of course that you are not
As for additional task lighting, I just run those off a separate switch so I don't have to worry about messing with the garage door to keep them on.
I was kind of hoping that I could use some sort of relay to act as a 3rd switch on the "task lighting" circuit and toggle them on when a car entered the garage.
Like others said, if you just don't want to walk to the switch to control them, motion detection lights are plentiful and reasonably priced these days.
I'll look into that. In the garage it may be ok, but I have not had good luck with motion sensor lights outdoors in the winter (they turn on randomly).