I was caught shoplifting items worth 175 $ (4 year ago), the cops were called, they arrested me , and had me finger printed and photographed at the station. I had my first offense.I paid 600 $ for penalty
I have green card , can I travel out of country and return back to US ? 4 years I am living with my family in New Jersey

You should be able to travel Was there a reason for the crime ?

You should be able to travel overseas and return to US without any problem. However, to apply for naturalization will require you to disclose all info which may effect your case status. Keep low profile and stay away from all legal problems.