Summarize:

Take photos of the car from all angles if possible. Ask a family member or friend with a camera to come and take photos if you can’t. Take photos of damage and area before you leave the scene or your car is moved for any reason other than clearing traffic. You need to file a claim that includes the photos of the damage to your car, the police report, and any information you collected on the pet (including its owner). You can call your insurance company’s claims representative or file online. The company will evaluate your car’s damage values and recommend one or more repair shops to use. This will require comprehensive coverage on most policies. Use one of the shops your insurance company covers so you get properly reimbursed. This most likely will mean body work if you hit a dog or a cat.
Look for damage on your car. Contact your insurance company. Take your car in for repairs.