Store in an open container at room temperature, in the shade. Whole heads last longer than slices.
Cut in half (from one end to the other) and the peel should come out easily.
Beginner recipes will usually call for chopped onions. Slices are also great for sandwitches. Whole onions or wedges are good if the goal is to extract the flavor and remove them later.
There is a technique that can save you time. Start with a peeled half of an onion (cut end to end), and make cuts towards the stem, but leave half an inch at the end so the onion stays in one piece. (Hint: Use the tip of the knife to start the cut, so that your knife doesn't slip too far forward). Once you're done, rotate 90 degrees and cut to cubes.
Note: Some people suggest also making length-wise cuts. Seems unnecessary, and dangerous since you're cutting towards your hand.
Note: Some people suggest cutting radially to break fewer cells. Supposedly it results in a smoother taste and less tears. I haven't noticed a difference so I'm not doing that.