> However for people who just check out with their credit cards
There are plenty of solutions by the various payment providers, e.g. https://stripe.com/docs/payments?payments=popular ranging from buttons that open a new webpage (on the payment provider domain) and return after success to iframes that look like your own website but securely transfer input to the payment provider's systems. Paypal does ask the user to login but I think that's optional. Stripe doesn't have user accounts so Stripe doesn't ask users to sign in.
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.