I love shopping at Publix and less than a year ago I signed up for a great program that they offer for babies, The Publix Baby Club
When you sign up for this club you get a free book "Caring for your Baby and Child" as well as several coupons for practical items in their store. Many of the coupons were for free items such as rubbing alcohol, hand sanitizer, cocoa butter lotion and more. They also send coupons for large discounts on items such as diapers, wipes and baby dishes. In addition, I received coupons for items that you just don't ever find coupons for like meat from their deli and other meats.
Before my daughter's 1st birthday I also received generous coupons for ice cream, cake and Mylar balloons. When I say generous I mean coupons that range anywhere from $1-$6. They also accept manufactures coupons and frequently have buy one get one free sales (where you don't have to get 2 things to get the discount).
Publix also has a program for preschoolers called the Preschool Pals Club. I have not yet participated in this program but if you visit the link you can view all of the things your preschooler can receive.
Another reason I love Publix - they have "car" shopping carts which keep my little one entertained (along with sanitizing wipes for the carts), free cookies for kids, free coffee for parents (though I've never tried it) and the service you get there is excellent.