I set this up a couple years ago but I seem to remember AWS walking me through the initial setup.
First you'll need to configure your domain(s) in SES. It requires you to set some DNS records to verify ownership. You'll also need to configure your SPF record(s) to allow email to be sent through SES. They provide you with all of this information.
Next, you'll need to configure SES credentials or it won't accept mail from your servers. From a security standpoint, if you have multiple SMTP servers I would give each a unique set of credentials but you can get away with one for simplicity.
I've got postfix configured on each of my VPS servers, plus and internal relay, to relay all mail through SES. To the best of my knowledge it's worked fine. I haven't had issues with mail getting dropped or flagged as SPAM.
There is a cost, but with my email volumes (which are admittedly low) it costs me 2-3 cents a month.