You should check to see what server side language (ASP, .NET, PHP) and database (MS Access, MSSQL, mysql) that your server supports. And then you can probably install your own shopping cart.

You might need an SSL cert as well and this will require a dedicated IP address if you wish to process credit cards on your website. You might also be able to use a shared SSL from Godaddy. A lot of payment gateways also will give you a secure web page to use to capture and process the credit card transaction.