If you are injured in an incident that was the other driver's fault should you go to your insurance company or to the other driver's insurance company to file a claim?

This would depend on which state you live in, and whether or not your required by state law to carry medical coverage (PIP) on your policy.

If you do carry PIP coverage, you would go through your own carrier for medical bills, even if you weren't at-fault for the accident.

If not, the other carrier would need to pick them up (assuming their driver was at-fault). You might also have a bodily injury claim against the other driver.

Keep in mind that if your state is a "no fault" state (ie, requires PIP medical coverage on auto policies), you don't get to choose to go to your own carrier or the other. Your own carrier MUST pick up your medical bills.

In Ontario you got to your own insurance company