Most prebuilt shopping carts have several options for payment gateways or even taking payments themselves (requires a secured server environment).
Payment gateways are just like paypal. AuthorizeNet, ChecktoGo, and SitePoint are just a few others. Search 'payment gateways' for a list. Research which one you want to go with. All have their pros and cons.
You'll probably want to start looking at the prebuilt ecommerce carts that you can install on your server/host and customize to fit your customers needs. Especially if you don't know much about php or asp.
If you are going to go this route in your career, I would suggest that you take a look at php and at least learn to recognize it's structure. All programming applications still output HTML and you should be able to recognize PHP codes (or what I call patterns) and output(the assembly of html) even if you don't know how to program php from scratch.
Most ecommerce platforms have the output code separated from the programming language by using a template system. This can usually be found in the template or theme folders.
If you have a really good base knowledge of HTML/CSS and a dash of Javascript, you should be able to figure out the rest of the template systems to customize the ecommerce carts to your customer's liking.
The main thing to remember that if you are just starting out with these systems, to take an existing template or theme and then replace the graphics and colors with your own. It's a fast-track to get things done.

Jade