Someone Hit My Car and I'm Not Hurt But My Car Is Damaged. Should I Get An Estimate or Just Call the Other Driver's Insurance Company?

insurance form other driver was at fault

Neither. I say you should first call your own insurance company if you have full coverage. Your company will move much faster and will generally treat you better. You will have to pay a deductible, but you should get that back later when your insurance company "subrogrates" back to the other driver's company.

If you don't have collision coverage insurance, then quickly call the other driver's company and report the claim. They will move slower, but ultimately, you should be able to get them to pay for the repairs and provide you with a rental car. There is no need to go get estimates on your own - the insurance company will send out an estimator for you, or will ask you to take your vehicle into a shop.

You have the right to pick your own body shop. But don't expect to get a big, fancy SUV for your rental; you usually are only authorized a small $25-per-day car.

You might also want to read: 

How Do I Get My Car Fixed After a Fort Worth Auto Accident?

How Do I Get a Rental Car After My Fort Worth Auto Accident?

Can I Make a Claim if I Was Hit By a Driver Who Fled the Accident? | Texas Lawyer FAQ