Upvote:2
If you only travel to Canada or Mexico or Caribbean islands for less than 30 days, you can re-enter as F-1 status using Automatic Revalidation and won't need an F-1 visa.
For any further travel, if you want to re-enter as F-1 status, you will need an F-1 visa. Since you don't already have an F-1 visa, you will need to apply for the visa at some US consulate. This consulate doesn't necessarily need to be in your home country -- you can try to apply for a visa at any US consulate, but the chances of success will likely be best in your home country.