Grab the wheel at the top and try to move it side to side in the frame. If there is any play, if there's any movement at all while the wheel is correctly mounted and quick release correctly tensioned and set, the hub bearings need adjustment.
If this is not the case, look at the rim/brake interface when spinning the wheel. If you notice that the rim gets closer to the brake surface, or in extreme instances, if the rim hits the brake pad at certain points, the wheel needs to be trued.
The pedals moving slightly while spinning the wheel -- there might be a bit of friction within the bearings of the freewheel/freehub which will go away over time.
If the gears are jumping when you do this, probably cable tension to the rear derailleur needs to be adjusted.
If you bought it new at a bike shop, they should adjust things correctly as part of a normal new bike follow-up service. If it's a used bike, pay for a tune up and mention the issues you listed here as concerns on your end.
I know next to nothing. I am frequently wrong.