How to automatically backup a MySQL/Amazon RDS database?
This is not possible using RDS snapshots.  AWS RDS Snapshots are "hidden" (though almost certainly in S3).  They are not accessible to you, the user, at all.  They can only be restored to RDS.

The only way to get data out from RDS MySQL instances is to export it using mysqldump (http://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/MySQL.Procedural.Importing.SmallExisting.html is a guide to importing, but works equally well for exporting), or by configuring a MySQL replica (http://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/MySQL.Procedural.Exporting.NonRDSRepl.html).