(Updated for Patch 10.0.5)
Feb 6, 2023: Added and refreshed Weakauras and compiled a list of addons below.
For general Priest play
This is a heads-up display I keep right under my character model. Helps me determine at a glance what’s off cooldown that I can cast (and I can see my feet in case I’m standing in anything particularly dangerous). Switched over to Luxthos for Dragonflight.

A bar that shows your Innervate duration. I position this right over my player model.

Dragonflight Consumables and Raid Buffs
Shows you what consumables or raid buffs you’re missing and if anyone in your party or raid is missing Power Word: Fortitude. Your raid leader will love you.

If you’re like me and are pugging dungeons or raids and don’t really know who to hit Power Infusion with, this will let you know when players are using their damage buffs. You can adjust the priority of classes. This is great if you just want to get it on cooldown and don’t really care to know who to give Power Infusion to in a pickup group. Or you could just give it to a tank and get yelled at.
Answered Prayers and Apotheosis Tracker Bar
Places a vertical bar that shows the time left in both Apotheosis or from the Answered Prayers talent. You’ll want to use this time cast the spells that reduce the cooldown on your Holy Words (like Prayer of Healing for Sanctify and Heal or Flash Heal for Serenity).

Places a glow on a raid frame if someone drops below 32% health and Power Word: Life is available.
Tracks your equipped trinkets. I like to position this above my player frame. Will highlight if you have an on-use ability (like Mote of Sanctification) or display stack counts (Flask of the Solemn Night).
For Dungeons
Displays general key information and timers. By Reloe.

Dragonflight Season 1 Dungeons
Contains bars, timers, and action responses to incoming abilities (Keep moving, spread, drop, etc). By Reloe.
M+ Explosive Orbs – Nameplate Glow
For Explosive week, allows for easy tracking of any explosive orbs. Shadow Word: Pain gets rid of them instantly.
Drops a moon on the mob with the highest health.
Spell CDs on Nameplate – DF Season 1
Shows cooldown of spells next to enemy nameplates for DF Season 1 Dungeons. Useful so you know if you need to Chastise or Psyhic Scream something.
TJS – Wise Mari Corrupted Geyser
For the Wise Mari encounter (first boss) in Temple of the Jade Serpent. This should cut down on the amount of deaths players have from mistiming the Corrupted Geyser when running through the water and getting bounced up.

For Raids
Ties into DBM and Bigwigs and makes a vertical timeline bar of upcoming abilities.
Vault of the Incarnates – Glows
Great for healers. Tracks key ability targets on players that would need immediate attention as glows on your raid frame.
Addons
Hands down the best bag addon made. Sorts all the stuff in my bags in to various categories like tradeskills, armor, consumables, quest items, and so forth.
After using DBM for over a decade, I decide to just switch to Bigwigs for this expansion.
I use Luxthos plater profile.