Like most have said here, Paypal is costly but it's a good start. It can also be the most economical, depending on how much business you do a month. If you only do a few hundred dollars' worth of online transactions a month, something like Paypal will cost you less. If you process over USD$1,000.00 each month, a merchant account may be a more economical solution. If you have a business account with Paypal (you can have two: one business account, one personal account), you will be able to accept credit cards without your customers having to sign up for Paypal.
Depending on where you do business, and how many customers you get from which regions, I would look into an alternative like 2checkout. I don't love them, and it's been years since I used them; but they are accepted in some places where Paypal is not.
For determining shipping...I have no idea. I've always ever sold virtual goods, so haven't had to fiddle with that functionality. That said, I'm sure Paypal has something to determine shipping - heck, they've been an Ebay preferred processor since before Ebay bought the company. Check their help documentation.
