You need to signup to a payment gateway. It's a service that allows you to process credit card payments. I know you don't want to use PayPAL to accept payments, but PayPAL offer two solutions:
PayPAL Payments - That is the button and people pay through PayPAL
PayPAL Payments Pro (Payment Gateway). That is when you use PHP or anything else to connect to PayPAL with the credit card information. The customer never knows you used PayPAL, and they never leave your site.
When choosing a payment gateway these are the four most important things you should consider:
How easy is it to implement (PayPAL has a lot of documentation)
How much is the monthly fee.
How much do they charge per transaction.
How much do they charge to setup an account.
There are other options, and one of the most common is Authorize.net, but the is more setting up and a larger deposit to open an account.
Most accounts will require a background and credit check.