For the fellas, this date is a mysonginists (did I spell that right?) dream due to the high likelihood of an impromptu wet t-shirt contest breaking out between the two of you. Plus, you might get your car washed. So that's a plus, right?
Participating in a task together is actually a good way to bond and create rapport. The more comfortable you are together, the easier it'll be to get to know each other.
Washing your/their car together is ridiculously easy, accomplishes a task that you might otherwise not enjoy doing on your own, and allows for you to get playful or even use some pseudo-social-science about why it's best to wash a car from top-down or back-to-front. You can actually just make up the answer as long as you sound confident about it!
So grab a bucket, sponges, some soap and a hose, and have a wet and wild time with your date.
There are definitely a few prime-times when it comes to washing your car. For instance, the Summer is far better than the Winter and afternoon is much easier than evening. Basically, this is a great mid-day activity during the warmer months of the year. It's even great to pair with some sort of hike or driving excursion where you may need to wash the car afterwards.
Obviously if either of you have a drive-way with hose access, this is ideal. However, back when I lived in an apartment, this wouldn't have been an option. If that's the situation you're in, there are often "do it yourself" or self-serve car wash places where you can pull into a stahl and clean your vehicle for a small fee. Consult Google and you'll probably find one that's reasonably close.
Otherwise, if there IS a driveway available to do your car-washing in, then this is probably the best place as you're not on any time constraint and have more follow-up free/cheap options at one of your homes than if you were at a self-serve car wash.