If I ask Amazon to send a book as a gift for my granddaughter, the "shipping address" is my granddaughter's address in Pennsylvania, where she lives. I want the book delivered to her, not to me. The shipping address is the place where the item is sent or delivered.
The "billing address" is my own street address here where I live. It is the place where the credit card company mails my credit card bills. And if I didn't pay :) it is the address that would be used when taking legal action against me.
Very often the shipping address and the billing address are the same.
"Shipping" can mean any kind of transportation of goods, not just over water in a ship.
"Bill" has a range of meanings all involving pieces of paper with amounts of money written on them. It often means "a request to pay money." "The electric company sends me an electric bill every month." "I don't like my job but it pays the bills" (i.e. I earn enough money that I am able to pay all the bills I get).