Genie Garage Door - Opens a little - then quits

That's where to start, see how much force it takes to lift the door by hand. If shouldn't feel too heavy. If it does, then the springs need to be adjusted or there could be a broken spring. If it doesn't feel heavy, then adjusting the up force on the unit is next.

My Genie opener runs for approx 1 foot, then shuts off. I have to unplug the opener from the ceiling, then plug it back in to get it to work again, and it does the same thing when i press the button. I've disconnected the door and it still runs for a second, then shuts off. Only the light bulbs can be turned on and off. Is my motor shot?

Common problem with all brands, mostly for one or more of the following reasons:

Broken spring that hasn't been noticed, dirt build-up in the roller track(s), dirt build-up in the roller wheel axles, travel limit control setting loosened and shifted, up direction force strength setting loosened and shifted. Suggest you disconnect the opener from the door (pull the "release" cord that usually dangles down from the spring loaded latch where the door hardware connects to the opener hardware) and lift the door manually. If the door is OK, movement should require not that much force and travel should be smooth for the entire way between fully closed and fully open. No squeeks or rattles should be heard, just the sound of the roller wheels moving easily along the tracks. If the door seems balanced and moves smoothly and quietly, the problem is more likely with the opener. If door motion isn't balanced, smooth and quiet, the problem is likely with the door and its hardware and not with the opener. Checking those things should help figure out what might and what is not likely to be causing the problem.