If you are getting fresh oysters, you want to make sure the shells are unbroken and tightly closed (cannot pry them apart with your fingers). If there are open shells, they should close promptly after tapping. They should be stored on ice, and no older than 4 days from harvesting for optimum quality. You can store them in the fridge once you get home, in a bowl covered in a damp tea towel. I would keep them in there for a maximum of 2 days before eating. If you store them in a tight container, they will die. If you are shucking them, they are easier to open if you chill them in the freezer for 1 hr +.
If you are collecting them yourself, discuss with the local fisherman about red tide, just to make sure there isn't any (some places have signs up year round warning of red tide, so it's not all that useful in the end).
Alternative to fresh oysters: smoked, frozen, and canned. Smoked are so delicious, most local fish shops will have some smoked oysters around, either fresh or frozen, or you can request some and they're usually happy to bring them in. Frozen oysters are pretty great, you can usually get them from fish shops for a really good deal (it's easier for them to store in the freezer than to keep them fresh and sell them in the time window). Canned oysters are a great alternative, look for a BPA free can. I love canned smoked oysters, occasionally get a strong craving for them and just eat them whole! Also great flavour to be added to spreads and put on veggies.
My boyfriend used to part-own an oyster farm, and we had an unlimited supply. Look around for recipes, and you'll find it does not really ever get boring- oyster stew, barbequed oysters, raw oysters, smoked oysters...so good!