In America, most supermarkets have bags, so you don't have to bring your own. However, most stores also encourage you to bring your own.
As far as who bags the groceries, that will depend on the store. Some places require you to bag your own. Most supermarkets bag them for you. Some places ask if you need help taking your bags to your car for you; some don't (but they will usually help you out if you ask for it, unless you're at a place like a Super Walmart or Super Target).
Most people pay with credit or debit cards, and you swipe it yourself.
Hope this helps! Are you coming to America anytime soon?