I am currently in OPT. I would like to know if I can travel outside the United States and hope to return by applying for a renewal of my F-1 student visa.
If you qualify for automatic revalidation, you can make short visits of up to 30 days to Canada or Mexico and return, so long as you have your I-20 (and I-94, if you have a paper one).
If you go to another country, or you don't have an I-20 endorsed for travel by your school, you will not be able to return without a new visa.
After you apply for a new visa, you also won't be able to return until the visa application has been processed and you have received your new visa. You cannot return to the United States while your visa application is pending. You will need the I-20 to renew your visa, so get it before you leave the United States.
This is a brief summary of the complete rules, which you can find at the ICE web site.