Wife parks too close to my white car. Would love to get rid of these door dings as her car is only 3 years old.
tl;dr: It depends on the ding. Try some cleaner wax, though: it won't make the problem worse.
Did you scratch down to the primer or just leave some paint behind?
If all you did was trade some paint between cars, you're going to need to polish it off. That said, if you're not comfortable with power polishing tools or abrasive polishes, I would counsel a cautious approach and shoot for incremental improvements.
CAVEAT: never wax or buff in direct sunlight. Misery awaits. Once upon a time, I put a whole set of swirl marks into my paint trying to get over-dried wax off in the sun.
So, here's what I used to do (before I bought the above-mentioned power tools):
The cleaner wax is doing a few things: it's abrading some of the mark loose with each application, suspending those loose particles in the wax and carrying them away when you buff the dry wax off. In addition, the remaining buffed wax softens the edges of any dent or scratch that you made in the wife's car (i.e., they look less bad). Finally, the optical properties of the wax will reduce the probability that a glancing light will catch a perfect specular reflection (i.e., remaining scratches are harder to see).
Note: I have a random orbital at home and I go straight to mild polishes in these situations. This is an investment and not a step that I'd recommend to a novice or those in a hurry to get some immediate gratification.