What would I need to get a part-time job at Kroger or Publix?

You need a work permit because you are under 18 and GA law now requires that for anyone under 18. You get that at your school or board of education and your parent or guardian needs to sign. http://www.dol.state.ga.us/pdf/forms/dol4111.pdf will give you complete info on this. Go to the store and ask for an application. Fill it out neatly and completely. Take with you names, addresses, and phone numbers of 3 people you can use as references. This might be a teacher or principal, pastor, youth leader, club leader, neighbor, but not relatives. Ask them if it is OK before you give their name that way they are prepared to say good things about you. Take a copy of your work permit. Take your driver's license or learners if you have it or other ID and your social security card.