Do I go to the hospital or doctor?

This depends on the severity of your injuries. If an ambulance is required for your injuries, you should take the EMT's advice and go to the hospital if necessary. If you're injured, but do not need emergency services, visit your doctor as soon as possible. Getting the injuries documented will go a long way towards getting you the settlement, or court decision you deserve.

Do I talk to the other party's insurance?

Only deal with your insurance. If you discuss things with the other party's insurance, they may try to get out of the responsibility of paying for your injuries. Getting an attorney to help handle the claims may also be beneficial to you in the long run.