There should be a small amount of detectable play in a properly adjusted hub when the wheel isn't in the bike. This will disappear when the QR is tightened up. If there's no play with the wheel out of the bike, the bearings will be too tight with the wheel in place, and ignoring this can damage the hub.
Judging the correct tightness when locking the cone against the locknut takes a bit of practice, and you may have to have several attempts before you get it right.
You should adjust the cone that's on the left side of the bike.
If the left locknut comes loose, the bearings loosen and you get a slightly wobbly wheel. The QR will stop it undoing very far.
If the right locknut comes loose, the cone can screw itself into the bearings and tighten them up so much the wheel won't go round without a lot of effort. This isn't good for the wheel, as well as making it hard to get home.