[mytailorisrich]

The topic you need to research is, among others, "PCI compliance".

The bottom line is that you do not want to handle payment cards' details.

With modern solutions like for instance Stripe's this is not a problem (apart from somewhat locking you in): they transparently handle cards' details and can store them so you can access that card later for further payments if needed. Customers need not be aware of anything or that you're using Stripe at all.