After my mother-in-law had a minor fender-bender, her minivan needed a new headlight, front bumper, and associated repairs. Here's how I saved money:
1) I purposefully avoided the repair shop recommended by the dealer as they have a reputation of high prices (and long wait times); rather, I tried a small shop recommended by neighbors. This shop quoted $1,723, which was understood to be less than the other shop. Since I don't know much about car repair, I would have normally accepted this quote just to be done with this business. However, empowered by a bit of Mustachian motivation, I took a little extra time to get a second estimate, which came in at $1,330, for an easy savings of $393.
2) As an added bonus, the second estimate was so complete that it included the parts numbers. The owner agreed that I could supply the parts on my own. I spent a little time on ebay and bought the required parts for $206, which brought the initial estimate from $1,330 to $758, for a savings of $366 ($572 less $206).
Total savings: at least $759.