Artichokes are easier than you might think.
Step 1: Rinse and trim
Find some good-looking artichokes. I usually get a bunch of the small cheap ones instead of the big expensive ones, because the leaves are usually younger and more delicate. Brown on the outer leaves is fine- it's caused by frost, which can help concentrate the flavor. Mold, and black or mushy spots, are to be avoided.
Wash and drain your artichokes, then chop the top layer of prickly spiny bits off. While trimming isn't absolutely necessary, the lack of spines will be appreciated when you peel the artichoke later. It can also help facilitate more even steaming.
Especially with the larger, coarser artichokes, some people prefer to use scissors to trim off the spike from lower leaves as well. I see this as another good argument for buying the smaller, younger, cheaper artichokes which rarely have spikes below the top leaves.