Spend the first few pulls to learn the tendencies of your tank. Figure out his/her Flash (for PLD) and Overpower (for WAR) timings. It's relatively safe to start healing after that. If your DPS gets hit don't panic, heal enough so the DPS don't get 1 shotted and let the tank work on grabbing the mobs back and then top off the DPS hp when you have a chance. If you pull aggro dont' panic either, healers are not that squishy in this game. Kite the mob back to your tank or u can fluid aura them. Communication with tank is key!

As long as you are not over healing, you shouldn't have to worry about pull aggro in main bosses unless they spawn adds. Most of your aggro should come from adds or trash mobs. Study the enmity gauge (the little vertical bar that appears on the party bar beside your class/job logo's when you target a mob on your enmity list). Also learn the colour representation of the mobs on your enmity list. If it's red the mob is on you, orange is be cautious.