Auto attack your opponents as much as you can without drawing creep aggro, try to get used to predicting their movement to get the most out of your Q, save your silence for ganks or all-ins, ONLY heal your adc if he is below 60% hp otherwise he could lifesteal up and you'd have wasted your heal.

The moment you hit 6 you need to notify your team, throughout the game from that point you will want to be watching your team mates health bars, if one person drops below 40% from poking then DON'T use it, save it for when your opponent seizes the opportunity and uses their summoners to attempt to secure the kill.

Example: --Ahri mid against zed--
Zed pokes Ahri to 40-50% hp;
Ahri plays more passively;
Zed flashes under the tower and ults Ahri igniting her (to secure the kill from deathmwark)
You ult and save her life whilst using up zeds summoners and ultimate.

Another good time to use your ultimate would be if a teammate is going to secure a kill without any assistance and you will get the 50% gold assist bonus for literally doing nothing (though this is greedy!)

In team fights you will want to stand towards the middle of your group so you can reach everyone with your W, use your q whomever you are most likely to hit (or the adc if you think you can hit them in the center) and spam your heal on everyone that needs it. Silence channeling effects (like malz ultimate) and such, you could use it to block exits since it has a root effect but you will need to master the more basic routines before you can do that effectively. Use your ultimate the moment your team start getting low (or if your adc is getting focused use it when he gets below 40% hp)
Heal in this priority (varies on health lost for targets):
#1 Most Fed
#2 ADC
#3 Mid laner
#4 Jungler (if he survives then you could secure objectives)
#5 Top Laner

Use these as a guideline, depending on where people are and what objectives are more contested you will want to change where you ward. These however are the core warding spots.