Payment is mostly done through https://www.paypal.com/.
It is secure and trust worthy BUT Ebay owns Paypal and they will get you coming and going with fees.
You can take money orders and personal checks if desired but most people want instant gratification and their product asap.
Shipping is your choice, UPS, USPS, FedEx, etc....you can log onto their websites to create get estimate shipping quotes.
I assume Ebay will have a FAQ's section you can look into for more info. But, first of all you will need to create a free Paypal account for people to pay you.