Leveled up Healers and Tanks first and now doing my DPS classes.

The most important thing, especially considering the number of new and inexperienced tanks there are atm, is to pay attention the the enmity indicator.

The enmity indicator is the small bar below your Class/Job-icon in the groupframe.

It will show the current enmity the groupmembers have for the targeted enemy.

Pay attention to not get your bar to 100%, as you will have all the enmity and the enemy will attack you.

Like others said, use your enmity reducing abilities if you have, and/or switch your target, if you see your next attack will most likely turn the enemy on you.

If you happen to get the enemies attention, like others said, do not run away from the tank and if the enemy is not likely to die in the next 2-3 hits: Stop doing damage till the tank has the enemy back!