will be able to cope with what you want to do, and there are several third-party payment processing companies which will handle your card payments for you on their own secure servers - for a fee.
Alternatively, you can use your PayPal business account to process card payments, but again there'll be a transaction fee, and again the payments will be processed on PayPal's own secure servers.
You can set all this up fairly easily at your end - PayPal will generate 'Pay now' button code which you can insert directly into your HTML pages, which takes your customer to the secure automatically. The payments are then transferred (less the fee) to your registered bank account.
All of the popular design applications (NetObjects Fusion, DreamWeaver, Expression Web, etc.) will easily cope with all this, and most of them (but particularly NetObjects fusion) have templates that you can use for your site pages.