Artichokes are actually giant blossoms. It's best to get ones that are short and fat. Long and fat is fine, but always take those that are fattest at the base. It can be five inches long or a foot long- you are only eating the very bottom part anyway. Better to have one with a six inch diameter than a three inch.2) Preparation for using the hearts
If you are serving them whole for company, get one with the best appearance possible, but if you are just going to use the hearts, don't worry if the leaves are discolored or a bit wilted or generally not very good looking. The heart will still taste wonderful. But of course go with the best looking ones unless the little ones are fresh-looking but the bigger ones are looking a bit ragged.
First, rip off all the little leaves sticking out from the stem and trim the stem off, leaving 3/4 of an inch or so. Next, using a sharp serrated knife (the cheap, microtoothed "ever sharp" blades they sell on TV are actually best for this), "saw" off the tops of the artichokes. This will generated a lot of refuse to chuck out, but it's a quick job. Leave about an inch and a half to two inches from the base. Place them in a steamer basket in a dutch oven or soup kettle with a couple of inches of water. Steam them for fifteen to twenty minutes.3) Preparation for Serving with leaves intact
When they are done, place them in a cold bath. Now is where the tricky work begins. With a spoon, scrape out the "choke" (the stringy, fibrous stuff in the middle). Don't scrape too hard or chuncks of the heart will come off with it. Quick, steady strokes should do the trick. After you've done it a couple of times it isn't any more difficult that peeling a potato.
If you don't mind wasting some good flesh, now just rip off the outer leaves and with some poultry shears, snip off the outer leaves to reveal the rest of the heart. Finally, snip off the rest of the stem. It's pretty simple.
Or, if you don't mind putting in some more time and energy to get more artichoke for your money and effort (I'm one of those guys who eats all the meat out of the lobster feet and picks through the insides to get every bite) you can scrape out the flimsy little leaves on the inside surrounding the heart (these puree wonderfully with all the great artichoke taste of the heart) and you can even scrape off the yummy bits from the bottom of the bigger leaves. If you are swift about it, you can quickly amass a small pile of goodies to puree into a sauce or to add to the overall puree, if that's what you plan to with the hearts. Even then it isn't hard, but it can be a bit time consuming if you feel like doing the job right without waste.
This is another matter. There are several ways they are commonly prepared; they can be steamed, baked, grilled, and/or stuffed with all kinds of good things. ou can serve them with drawn butter or just about any kind of sauce you like. I'm not going to post recipes here, because there are so many of them out there and they are easy to find. Bear in mind that artichokes discolor quickly when they are cut, so you'll want to to have some lemon juice handy to dip them in to keep them from looking rather yucky.
You can freeze the hearts for later use, if you like, but I've found that these things are pretty perishable and will go bad in just a couple of days. And when these babies go bad, they go REALLY bad- like broccoli or beans. And then you've wasted your time, money, and you have no yummy artichoke hearts to eat. Nobody wants that.
So you can slice them and put them on pizza, you can put thm on salads, your can serve them whole in any sauce of your choosing, you can serve them on the side next to your entree. You can put them in lasagna, ravioli, calzones, etc. You can puree them to make soup, or to make a wonderful, nutty sauce for fish or chicken. You can marinate them and serve them as an antipasto or an appetizer. They are really great, and very underused.