Don't know about the cost. You can get in touch with ESRB thru their website
https://www.esrb.orgThey will ask you to mail them a disk with playable build of your game along with other material. You may need to send walkthrough footage of your game, which can be a pain to record sometimes.
If you are aiming at a "E for everyone" rating, be very careful about the content of your game as they won't let anything slide. For instance, I produced a PS3 game where we had to remove a single instance of a glass of wine to avoid getting an "alcohol" mention next to the ESRB label.
It can take a few weeks too so make sure you plan accordingly.