No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
You get 5x points for booking directly with airlines OR by booking through AMEX travel (online or by phone.) Personally, I tend to book directly with the airline as the fare I'm flying is the same price and Delta's website is easy to use.
I haven't used AMEX travel in a couple of years, but I seem to remember you can input your skymiles number during the booking process.
I would also book this directly with Delta. Still get the 5x and easy to input everyone's skymile numbers.
If you're using the regualr Amex Platinum card then it's best to just book directly with Delta. Won't really make a difference either way, but at least you'd go through Delta directly so if ever there was an issue, Delta would be the main contact.
I only book through Amex Travel if I'm redeeming Membership Reward points. They do let you put in your Skymiles number and you still get credit through them as well.
Book directly with Delta. So much easier to deal with changes and problems when there is no third party (i.e., Amex).