I use a simple digital stick cooking thermometer, round ten bux at any big box store. You can certainly mount a thermometer, but be careful, with a handheld you can easily move it around the mash and check for consistent temps....KISS
So a question on the process. How much would I shoot for over the strike temp? Then add the water to the cooler and stir in grain. Once I hit my temp leave it for the mash time or check it periodically?
Add the water to your cooler and let it sit for a few minutes to preheat. Add the grain, stir to break up any clumps, and check the temp. It should be right around your target temp. Add a little cold/hot water to adjust the temp to hit your target. I personally don't check after I hit my temp. Any opening of the lid will drop the temp.
Take this mashing time to double check your brew-day list, heat up your sparge water, and have a homebrew. Also, relax.