You can do everything right and still blow a tire. I did, in a hilly LA neighborhood, in the dark, when I backed my car into a pointy chunk of fractured concrete. BOOM! Dead tire.
Luckily, I had everything I needed in the car to make changing the tire easier.
I had a flashlight — one whose batteries I often checked. And I had EXTRA batteries, just in case.
I also have a good pair of work gloves, to protect my hands.
I also had a rain jacket, although it wasn't raining so I didn't need to use it.
And it doesn't hurt to have an old jacket or shirt to prevent yourself from getting grimy. A rag or two is an equally good idea.
This last piece is important. You should have in your car what you would need to change a tire in the worst possible condition for where you live. That means a warm coat, hat, and gloves if your neck of the woods is subject to winter weather.
It's also helpful to have warning flares on hand, should the flat or blowout occur on a highway.