All four covenants are VERY close for Holy Priest, none of them bringing anything insane unfortunately.
Ultimately, your covenant choice will not make a big difference in your throughput for holy. Necrolord and Kyrian are both close in throughput (2-4% of your overall healing), whereas Night Fae and Venthyr do not add any throughput, but may be better for other aspects of the game if you for instance want to do PVP, playing Discipline or playing Shadow.
The Kyrian covenant ability, Boon of the Ascended offers a 10s window where you have a high single target nuke that heals in Ascended Blast and a decent, spammable ability in Ascended Nova which heals around you for a small amount.
After the 10s window, you release a decent sized explosion of healing and damage around you. The spell power value of Blast is high for an instant cast and neither Ascended Blast or Ascended Nova costs any mana, so you have a 10 seconds time window of spending 0 mana.
The Kyrian covenant ability gives you a 3 charge potion that does not share CD with healthstone or healthpot, heals you for 15% of your max hp and removes diseases, poisons, curses and bleeds from you. This can remove some very powerful debuffs in Mythic+ or raids.
The Necrolord ability, Unholy Nova, is very easy to use in any environment and will provide a short soft-CD every minute that will heal and deal damage.
Soulbinds and Conduits
Kyrian: Kleia & Pelagos
Both Pelagos and Kleia are good choices at the moment.
Pelagos will let you put 2 Potency conduits, and his key ability: Combat Meditation offers you a large amount of mastery every time you activate your Boon of the Ascended. It has an extra mechanic, where you spawn orbs around you and walking into those extends the mastery buff by 3 seconds per orb. This is a very strong trait for both Mythic+ and raiding, since generally, you will be using your Boon of the Ascended on high damage intake moments, such as the last 10-15 seconds you are fighting the Prideful affix, making mastery very valuable.
Kleia will let you put only one Potency, because the Pointed Courage trait is worth the taking, being better than your second Potency conduit. It will give you +5% crit chance, which is a good stat for Holy as long as you have allies and/or enemies within 8y (which is very likely on most fights), and synergizes well with Valiant Strikes, even if it was nerfed since the early beta.
Potency: Resonant Words - this potency conduit offers you a very large Heal or Flash Heal after every Holy Word. In both raiding and Mythic+ this conduit will give you a very powerful single-target heal. Using the effect from Resonant Words is most effective on Heal, but do not be afraid to use it on a Flash Heal if you deem it necessary.
Once you unlock a second potency slot, you will likely want Focused Mending, which offers a small healing increase throughout the encounter.
Holy Oration will be better than Focused Mending, but only if it gives you a Holy Word: Salvation that you would not be able to cast otherwise, it is, however, going to be a risk that we do not recommend taking.
Endurance: Translucent Image - Reduces your damage taken for a short duration after pressing Fade. Gives Holy Priest just a slight increase to their survivability, which is much appreciated for a low mobility class without impressive defensives.
Legendaries for raiding
Flash Concentration is the default choice, but it will change your gameplay a bit. You need to stack 5x Flash Heal, before switching to an efficient Heal spam. Keep in mind that you must refresh the buff by using a Flash Heal before the 20 seconds expires. Luckily, this is an easy task thanks to the Surge of Light procs.
Harmonious Apparatus is another strong legendary, much more baseline; this one won’t affect your gameplay too much, as long as you keep spamming your low-CDs spells when available (Prayer of Mending, Circle of Healing for the healing-part), and granting you a slight DPS increase too if needed.
X'anshi, Return of Archbishop Benedictus is the “progress”/”cheese mechanic” legendary; after Spirit of Redemption expires, you will revive up to 100% health, based on your effectiveness during Spirit of Redemption. After reviving, you cannot benefit from Spirit of Redemption for 10 minutes.
X’anshi was a fan favourite in Legion when it was first introduced, and to the joy of holy priest players, it has returned in Shadowlands. Once per encounter, you return back to life after dying without consuming a combat res. It is great for progress where mistakes do happen. It is also going to allow you to keep pulls going for longer and allowing your guild to progress faster.
If you’re very confident in the encounter and you can find a good spot, dying on purpose to trigger Spirit of Redemption can allow you to produce a lot of healing with no mana constraints.
Legendaries for Mythic+
Twins of the Sun Priestess Power Infusion also grants you 100% of its effects when used on an ally.
In Mythic+, your optimal play is usually giving Power Infusion to a DPS player, generally, this will be on larger packs where Power Infusion boosting you as well will have some benefit. Even during boss fights, as mentioned previously Holy Priest has the ability to do a lot of damage in Shadowlands and this legendary will allow you to help your group with even more damage.