Remember you're a back liner and don't run ahead like I see a lot of Anas do. She does damage, yes, but you should be like 90% heals, 10% damage (as a solo healer, at least. If you have a Mercy with you, you can probably get away with more DPS). Also get used to projectile speeds because you're going to be doing hip-fire a lot, and not a ton of scoping. Her scope is nice, but it's usually only really useful when someone is far off from you.
Remember that your grenade is not only for healing your group, but that it denies the enemy healing for a few seconds.
Learn to practice your sleep dart. It'll be a pain at first, I promise, so don't get frustrated if you don't land a lot of them. I personally save my sleeps for either the enemy healer (usually Mercy), but mostly for when an enemy ults. (My favorite moments are landing it on an ulting Genji, or a charging Reinhardt.)
I also recommend trying to keep higher ground than your teammates, if you can, but this isn't always a viable/safe option.