So far as size goes, it depends entirely on your needs. Like Jason said, if you're cooking for your entire family, 12" is probably best. But it depends on what you're doing with it.
Important tip: a heavy, thick bottom to the skillet is important. A thick bottom will convect the heat evenly across the pan, whereas a thin bottomed pan will have hot spots, which leads to burnt food. The thick bottomed pans are more expensive, but it's worth it, if you look at it as an investment.
So far as cleaning and care, always hand wash in a mild detergent. Don't use steel wool, it'll scratch the bottom of the pan and that'll lead to sticking. Nylon pads, dishcloths or sponges only.
Barkeeper's Friend is extremely helpful, but not necessary. You can find it online, or in most specialty cook shops.
Make sure you dry it thoroughly. Mineral deposits from drying water probably won't do much damage, but I like the nice shine of well cleaned stainless steel.