How do I know if my cone/cap/bearings need to be adjusted?
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.