I think everyone should really level a tank, a healer and a dps - just to see how those three roles play and see things from a different perspective.
A good tank is watching what is going on and will often determine the rate of pulling based on what they see. Watching to see if the healer is keeping up well with heals, or if they are struggling. How quickly is the dps downing the mobs. The tank may or may not be familiar with the dungeon. Also, the tank may be very comfortable with their ability at crowd control, but they may not be comfortable either. (Also a good tank doesn't point mobs at the dps or healer and they also don't play spin the mob.)
A good healer is watching what is going on with the party. And (imo) will keep everyone's health up and on downtime do some dps when it's safe to do so. They'll let the tank know that they can handle big pulls if the tank is willing. They run a stray mob that is hitting them to the tank so the tank can more easily pick it up. They don't yoink the tank into more mobs.
A good dps will watch what is going on around them and with the party. They will down the mobs as fast as they can using aoe abilities on large pulls. If they are ranged and pull hate of some mob, they run to the tank so the tank can more easily pull it off. If the healer is getting pegged by a mob, then down that mob fast so it doesn't take out the healer. Don't run ahead and pull more mobs intentionally because you think the pace is too slow (say something in chat that we can take more).
Having played all three roles I have a better understanding of each and what they entail. I also know what annoys me as a tank, a healer and a dps and I try not to do those things to others because... they are annoying.