If I grab my [suspension] forks tightly and push against the wheel sideways, there is a slight amount (few millimeters on 26" wheel) of play/wobble from the axle/hub. The QR is done up tightly. Is this normal or does it require a hub service?
If you can feel any play ("rattling" sensation) at all in the wheel bearings, headset bearings, or crank bearings, then they need to be adjusted at the very least (and may need rebuilding). (But please do make a distinction between movement due to looseness in the bearings and simple flexing of the wheel or other parts because you're pressing on it so hard.)
Adjusting is something you can do yourself with a few inexpensive tools, a little mechanical ability, and some practice, but you won't get it right the first several times you try, so best go to your local bike shop unless you really want to learn how to do it yourself.
Standard loose-ball wheel bearings need adjustment every 3000 miles or so, and should be rebuilt (cleaned, new grease, and possibly new balls) roughly every 10 thousand (or more often, if subjected to severe conditions).