Let's say, for the sake of discussion, my client does not want to use the Paypal api for transactions. I have already made a shopping cart. The user submits the items on the shopping cart and their credit card number.
Assuming I'm not using any api, how would I then charge the user's credit card for the items on the shopping cart? I understand that it is complex, just wondering what the general process is like and how to get started.
You're going to need to trigger a bank transaction, so obviously you're going to need to use an API at some level. I think the way to go would be to first learn in some detail how at least one of the popular commercial (by definition?) payment card processing services works, e.g.:
The next step would be to design the "payment processing" subsystem of your PHP application/library around the "backend workflow" (perhaps using a 3rd party PHP library, if one is available), and do some testing.
Then I would plan for a security audit of your code & systems before you actually use it to handle payment card information on behalf of a customer/3rd party.