Is it important to mix up the different types of grains before they go into the mash tun or is it okay to just dump them in separately?
You definitely want to mix them at some point, either before adding the hot liquor, or stir them while adding them to the strike water.
If you don't get them well-mixed, then the grains that don't have enough diastatic power to self-convert (most specialty malts) might not get in contact with base malt. The result is that the enzymes from the base malts would not break down the starches in the specialty malts, and you would end up with a beer that is lower in alcohol and heavy on starches. This could be a particular problem in a beer with a large percentage of the grain bill coming from ingredients that cannot self-convert, like a wheat beer.
So it doesn't matter if you mix them before you mash in, just that you mix them once they're wet, and it's a good idea to stir the mash periodically, to ensure that there is no clumping and that all grains have access to enzymes.