Loose Hub?
 

My mechanic showed me that I had some slight wheel play on my front wheels. While on the rack, he had me grab the top and bottom of the wheel and showed me a bit of play. I guess (if I understood correctly), it is slightly loose at the hub.

Is this something I can tighten up? If so, how?

Are the wheel bearings adjusted correctly ?

I'm not sure. Let's say they are not. How do I adjust them?

Do a search.
I think that's another topic that has been discussed to death here. ;)

Yes,

I did. Is it as simple as removing the cap.. loosening the allen screw and tightening the hub screw a bit until no more play?

Yes.
DO NOT overtighten.

The ball joints could be bad.

Every tapered roller bearing front hubs I've tried the north-south shake on has had a play I can feel. I don't think this is a valid test. Maybe your mechanic is used to FWD cars with pre-loaded ball-bearing hubs?

I suggest you follow the advice at this site for proper testing and adjustment.

Steve