Anduin is a very popular healer who brings a fair amount of utility and playmaking potential in a team; however, Anduin is also very overrated and most people underperform on the hero, either by picking him at the wrong times, or by playing him incorrectly depending on the situation.

Anduin's healing output is not great, unlike what most people seem to believe. On the other hand, his Leap of Faith can really help pull allies out of situations there would otherwise die from, but do not be too eager to pull an ally from danger either, as very often you will have melee characters ready to use their own escape tool, or actually wanting to make a play, and you pulling them out will mess it up for them.

Another reason Anduin is picked is for his Lightbomb ability, which is a great combo with many heroes who can carry the bomb efficiently like Genji Tracer Zeratul Leoric, or simply to extend a CC chain with Johanna after she uses Blessed Shield for instance.

Otherwise, Anduin's main source of healing, Flash Heal, has a slight delay which hurts Anduin's performance against burst damage. Another thing to note is that, against teams that hard engage and dive, Leap of Faith will tremendously lose value.

This build is great for higher level of play as it mitigates Anduin's weaknesses to burst damage and dive compositions.

At Level 1, Power Word: Shield grants a shield to allies when using Divine Star, making it a great option to mitigate burst before the delayed heals from both Flash Heal and Divine Star can go off. It also grants Anduin a shield when hitting enemies, granting him extra safety when healing allies.

At Level 4, Piercing Light will make Chastise root an additional target. This is a defensive talent that can shut down multiple enemies that dive near you. The spell power increase is not a reason to pick the talent, it is secondary.
In case you do not find yourself against multiple enemies diving, Moral Compass should be picked instead, as it will give more value over the course of the game; the increase in AA range will make it safer for Anduin to proc Pursued by Grace, while also giving free procs of it with the 2 basic attacks fired at the tip of Divine Star, not to mention the extra damage.
If you aren't sure which talent to pick, go for Moral Compass, it is consistent value.

At Level 7, Desperate Prayer is once again an additional tool to help Anduin deal with burst damage. Note that it can be used on Anduin himself, but generally you want to use it on allies if possible.

At Level 10, Lightbomb is the default pick. It is a great combo with many heroes who can carry the bomb efficiently like Genji Tracer Zeratul Leoric, or simply to extend a CC chain with Johanna after she uses Blessed Shield for instance. Lightbomb can also be used defensively, as after the delay it will grant a shield to the target carrying it for every enemy hero hit. It is more reliable to use it offensively though.
Holy Word: Salvation is a great ultimate ability as well, but it is easily interrupted at higher levels of play. If you find that the enemy team somehow lacks the interrupts, picking this ultimate can truly win fights, but if you are getting interrupted consistently towards the beginning of the cast, you should not have picked it (it is fine if you only get interrupted after 2s out of the 3s, you still got good value from it.)

At Level 13, Speed of the Pious is the default pick, granting Anduin 30% movement speed (mount speed) almost on demand, but also reducing the cooldown of Divine Star for each hero healed by it. This greatly increases Anduin's ability to escape, but also boosts his overall healing output.

At Level 16, Glyph of Faith is very often the pick at higher levels, granting extra safety with a second charge of Leap of Faith.
Nonetheless, I recommend Inner Focus in general, for various reasons. First, if your team is getting dived, Leap of Faith loses value and is not saving anyone, thus a second charge doesn't bring much. Second, if you are solo queue, you will generally prefer getting and extra layer of healing, and this talent helps a lot to mitigate burst. And last, very often you will be on the offensive and need healing to stay on the offensive! Inner Focus allows you to do all this, for instance healing your Muradin while he is in the fray of battle so he can survive just long enough for your team to get those kills (now imagine if you pulled Muradin out instead... You would simply mess it all up and either lose the fight or retreat from a winning position.)

At Level 20, Censure is the default pick, it grants much more impactful CC with the extra stun and root duration, and in case you picked Piercing Light it can get even more value.
Note that planning to pick Censure is not a reason to pick Piercing Light on its own, as you may be playing without a Level 4 talent for the rest of the game, and that is not acceptable.
In case Holy Word: Salvation was picked, the Light of Stormwind upgrade may be game-winning in itself.
Similarly, Inner Fire granting 50 Armor to Lightbomb's target can truly save them, but you should mostly pick it in case you are using this defensively (note that the target gets 50 Armor instantly, before the explosion happens.)

This build offers a lot of sustained healing and is quite good for Storm League, or if the enemy is lacking burst damage.

At Level 1, Lightwell will heal nearby allied heroes 15 times. This active ability has a cooldown of 75 seconds, but every time Anduin heals a hero, the cooldown is reduced by 1 second.
The very interesting thing about Lightwell is that it gets cooldown reduction from EVERYTHING.
The Lightwell itself reduces its cooldown, Anduin's abilities reduce its cooldown, Anduin's basic attacks procing Pursued by Grace reduce its cooldown, if someone grants Anduin lifesteal with Vampiric Aura or Bloodlust that lifesteal will reduce the lightwell's cooldown, Regeneration Globes reduce its cooldown, the Healing Fountain reduce its cooldown, and finally the Renew and Varian's Legacy talents reduce its cooldown.

At Level 4, Moral Compass will allow Anduin to auto-attack more safely to proc Pursued by Grace, with a bonus on the 2 basic attacks fired at the tip of Divine Star, and all of this will of course reduce the cooldown of Lightwell.

At Level 7, Desperate Prayer is still recommended to be able to mitigate burst damage, but Binding Heal will have more synergy with Lightwell if Anduin has taken a bit of damage.

At Level 10, the Lightbomb vs. Holy Word: Salvation decision remains the same as in the previous build.

At Level 13, Speed of the Pious remains the safest and more consistent option, but Push Forward! has good synergy with the build in case you find it safe enough to constantly attack enemy heroes.

At Level 16, Renew gives the best synergy with Lightwell, but if you feel like you need more instant healing Inner Focus is still an option, similarly with Glyph of Faith if you really think you need it.

At Level 20, Varian's Legacy has great synergy with the Lightwell and enables Anduin to attack more as he will self-heal more when doing so.
It is still possible to pick other talents in case the situation requires it, but this talent is the one that completes the build.