Since I direct fire my mash tun to heat the water, I add the grain to the mash tun (would be kind of difficult the other way). Depending on how thick you're mashing, you might want to add about half the grist, stir/mix well, then add the balance before starting the timer. I've found that to make dough-balls easier to remove.
Keep in mind, there are several ways to actual mash, or to get to the saccharification stage. Most of us find a method that works really well for our hardware set (or current hardware set) and keep going that way.
When I was using a cooler for my mash tun, I would always add the water to the mash tun first, then add the grain. Worked really well for me then too.