Should I go through my auto insurance or the other driver’s auto insurance to repair my car?
If you have collision coverage under your own auto insurance policy, and the other driver's insurance company is accepting fault, then you can go through either your insurance or the other driver's insurance on the repair of your vehicle.
If you go through your insurance, you will have to pay your deductible, and then get it reimbursed from the other driver's insurance company.
You do not need to worry that going through your insurance will cause your premiums to go up. As long as the accident was not your fault, your rates won't rise. Your insurance company will subrogate (seek reimbursement) from the at-fault driver or his/her insurance company.
If the other driver's insurance company is still investigating who was at fault for the accident, then it will not yet agree to pay for the repair of your vehicle. In that case, rather than wait out the delay until liability is resolved, you will probably want to go through your insurance if you have collision coverage.