Three basic ways with artichokes:
Steam: cut off the top 2-3 centimeters and the stalk so it fits in your pot (once steamed, the white/yellow center of the stalks are yummy). Add water to the bottom of your pot (don't let it boil dry) and boil/steam for 25-35 mins until tender. Let cool on a plate then eat by pulling each leaf off and eating the bottom 1/3 -1/2 of each leaf. Dip in oil/lemon/salt/garlic/mayo, or a combo of any of these. When you get down to the tiny leaves, rip then off and you'll see something more fuzzy than leafy. Gently teaspoon the fuzz away and peel off the darker green of the stem. Eat the rest with gusto.
Click here for a link to a website with pictures.
Stuff and bake: you can trim and stuff with breadcrumbs, garlic and or meat then bake or broil. Here is great recipe; you could substitute the peas and beans with something else from the box, perhaps green beans, collard greens, carrots or kohlrabi and kohlrabi tops (add any greens 10-15mins from the end of cooking).
Slice and fry: One of our members, Michelangelo (the name says it all), told me to chop off all the darker green bits, peeling the outer leaves off and the outer base off. Then slice it up cut off any prickly bits in the centre and fry it up with garlic olive oil lemon rind. Boil up some pasta, add cheese, fresh herbs and artichoke. I usually add other things like sour cream, greens, olives and/or onion. Be creative!