How do I know if my cone/cap/bearings need to be adjusted? I have a custom road bike with (mostly) Shimano parts.  I love to ride but I'm not so strong on the technical side.

I have quick-release wheels; I regularly (3-4 times a year) put the bike in a bag and travel by train to a cycling destination.

When I did so, upon re-assembling my bike, I noticed there was a lot of "play" in the rear wheel left-to-right.  It rolls smoothly, but there is an intermittent....feeling.... as the wheel rotates when I am riding it.

The feeling is a feeling of some friction, but not enough to actually retard the forward motion of the bike or anything, just enough to allow my feet to feel it in the pedals.  It appears to happen once a rotation, as if I am going fast enough, I can't feel it anymore, but when I am going slow, there's certainly a pace to it.

I've heard anecdotally from a friend I may need to adjust my cone &amp; bearings, as they may have become too loose.  Does anyone have any advice on how to do this, or things I should watch out for?  I'm trying as much as is possible not to ride until I sort this, but since I commute by bike, it's urgent.
Pick the bike up so the problem wheel is off the ground.  Grasp the tire and push it back and forth (left/right relative to the frame).  If you can get any perceptible motion then the bearings are loose and need adjustment.

While the wheel is off the ground, spin it and watch the space between the rim and the brake block (assuming you don't have disk brakes).  The space should be nearly constant, varying no more than about 1/2 mm as the wheel rotates.  (Check this on both sides, since a dinged rim is also a possibility.) If it varies more than 1/2 mm or so the wheel is out of true and needs to be trued.

Both tasks -- adjusting bearings and truing the wheel -- are things that a competent basement mechanic can accomplish, but not particularly suited for a newbie, so in either case it needs to go to a bike shop.