You are allowed a 30 day grace period to travel before AND after your program.
Below is more information:
Travel Grace Period BEFORE your program:
You may arrive up to 30 days before the start date of your program as shown on the DS-2019. The Department of State explains the 30-day grace period before the start of your program here: http://j1visa.state.gov/participants/common-questions/
Travel Grace Period AFTER your program:
Observing the “Grace Period” After you have successfully completed your J-1 program, you are permitted to remain in the U.S. for another 30 days. During this 30-day period, which is often referred to as the “Grace Period,” you are under the jurisdiction of USCIS and no longer entitled to train or intern at your Host Company. The USCIS grants this period to allow you to settle your affairs and to prepare for your return to your home country. You may neither continue nor complete your program nor work during this 30-day period. Although you may travel inside the U.S., we recommended that you do not travel beyond the borders of the U.S., as you will not be permitted re-entry on your expired J-1 Training/Internship Visa. For your own safety, we also suggest you purchase health insurance for this period either independently or directly through your Sponsor organization.
It is important to remember the following:
You must contact your insurance provider to extend your coverage. Do not be in the U.S. without insurance! Your sponsorship ends on the date of the DS-2019. You should not travel outside the continental U.S. during the grace period; you will not be permitted reentry. If you are terminated from your J-1 program, you are expected to depart the U.S. immediately. You will not be entitled to the 30-day travel grace period.