I usually read them and watch them around the same time. Personally I prefer to read first, then watch, but maybe an act at a time. YouTube has productions of many of the plays up, but the actors and production value is often high-school level. Unfortunately there aren't many chances to see these with good actors unless you just happen to be in the right city in the right month, then you'll get to see Hamlet or Macbeth. You will not see Twelfth Night or Winters tale this way.

If reading, finding a copy with notes will be very helpful. I am a fan of the Folgers Library ones. Every page of Shakespeare is mirrored with a page of notes, which is insanely helpful