I buy them for myself a couple times a year (my wife doesn't hate them, but she doesn't crave them either). I find they last about 2-3 weeks in the refrigerator. After that, they turn black and while still edible, don't look good at all.
I have a pressure cooker to cook them (since they really do best when cooked at a very high temperature). I basically steam the artichoke in the pressure cooker by putting it in a bowl that floats on a 1/2 inch of water. Turn it on, wait 20 minutes, and you have a perfectly cooked artichoke. If you don't have a pressure cooker, any other sort of steamer will give you a great artichoke, but you might need to cook it ~30 minutes to get really tender leaves. Or you can just boil the thing in water (40 minutes or so), but it will be a little less tender.
Some people take the time and effort to trim the spiky ends, but since you don't eat those ends, I say "why bother". I've seen lots of recipies like Sepen's and always wanted to try them, but they always look like they put flavoring on the part of the artichoke that you don't eat, so again I haven't taken the effort. You can do it, and it probably is great, but certainly isn't necessary.
Artichokes are really good, but are missing something to be a really good eat. My favorite thing is to dip them in mayonase. I've seen butter used as well if you don't like mayo.
Sepen was right about how to eat it. The inside bottom section of each leaf is the best. I suppose you could eat the whole leaf, but that will be very hard to chew. Just eat the bottom part, and the scrape the rest of the inside with your teeth. When you see the small purple leaves, it gets too annoying to scrape each one, and they are so small, I just toss them out. Underneath is some hairy looking things - called the choke. It got that name for a reason. They are so annoying to eat that many people start choking on them (where the artiCHOKE got its name). Thus take a knife and cut them off. Underneath is the heart which is all edible and many people think it is the best part.
I choose to not eat the heart plain, but chop it up, add some frozen spinich, and some parmesan cheese. Cook it in the oven and you'll have a really good artichoke spinach chip dip.
