Here's my take on monks.
The Big 6 monk technologies at the monastery that actually matter i.e. they enhance the offensive capabilities of your monks.
Castle Age
Sanctity - +15 hit points, +20 for Aztecs (120 gold)
Redemption - Convert buildings and siege (475 gold)
Atonement - Convert monks (325 gold)
Imperial Age
Block Printing - Increase conversion range from 9 to 12 (200 gold)
Illumination - 50% faster reload (120 gold)
Theocracy - Only one monk reloads when performing group conversions (200 gold)
The order in which you research Sanctity, Redemption or Atonement isn't, in my opinion, that important and can vary.
In Imperial Age these upgrades should be researched in this order:
1. Block Printing is the most powerful Imperial Age upgrade. It's okay if you cannot reload your faith faster or the entire group of say 10 monks has to reload faith after a group conversion, but to be restricted to only 9 range is really dangerous at a time when when crossbows and skirmishers that get bracer will go up to 8 range, onagers have 8 range without siege engineers (9 if they do research it), bombard cannons have 12 range as standard, and melee units all get a little bit quicker in imp. Monks will need more time in order to convert things, and +3 range buys them that time.
2. Illumination should be researched second as it's 80 gold cheaper than theocracy and improves the performance of all your monks as individuals, which more important than their performance as a group.
3. Theocracy is a brilliant technology to research but unless you've already researched a minimum of block printing and illumination and you have at least 10 monks in play, then save your gold.
Monk Rankings
Tier 1
1. Aztecs
2. Teutons
3. Byzantines
4. Saracens
5. Spanish
6. Japanese
----
Tier 2
7. Chinese
8. Turks
----
Tier 3
9. Mayans
10. Britons and Franks
-----------------------
The top 8 civs are the only civs in the game with Redemption. They all have Atonement as well.
1. Aztecs
They have the all the technologies available, the +5 hit point bonus per technology researched, monks created 15% faster and the relic gold generation team bonus. They will probably end up on 75 hit points, because Faith and Heresy are too expensive, Herbal Medicine is useless and Fervor is bugged. In my experience the benefit you're most likely to get from the hit point bonus is being able to survive a direct mangonel shot. This alone is amazing. Large enough groups of Light Cavalry, Eagle Warriors and crossbows/arbalest/elite skirms will still make very short work of Aztec monks as even with more hit points they are still very slow and still have no armour of any kind.
2. Teutons
They have all the technologies available apart from Herbal Medicine (which is irrelevant). Their monks take 50% longer to be converted thanks to their team bonus. It's as if the entire civ has Faith researched from the start of the Dark Age. I think you could argue that because of this, they are the best civ in conversion wars. They also have a healing range of 8 instead of 4. That means healing range and conversion range are almost the equal. Now I understand how their team bonus works, I'd be more worried about using monks against them than I would against the Aztecs. Will we see the Teutons in MOA3? None of the settings for each game push people toward picking the Teutons.
3. Byzantines
They have all the technologies available apart from Herbal Medicine (which is irrelevant). Team bonus of 50% faster healing is good but not as good as the combination of the Teutons team bonus and healing range. Their fast imp capability may mean they are the first to get Block Printing, Illumination and Theocracy.
4. Saracens
They have the all the technologies available but have no monk related bonuses. Their Market bonus can help get you to Castle age faster and get the more monks out by using your market bonus to sell stone as seen in this game - https://www.youtube.com/watch?v=GJiZhfBg484
5. Spanish
They have the all the technologies available but have no monk related bonuses. There's the very slight potential advantage that their monasteries are built 30% faster so you could maybe grab the relics marginally quicker, but getting monks out a little earlier than other people who start building monasteries at the same time.
6. Japanese
They have all the technologies available apart from Heresy and they have no monk related bonuses. Heresy is rarely researched because it is so expensive. Even without Heresy, they are still a monk powerhouse.
7. Chinese
Exactly the same as the Japanese but they also lack Block Printing. Very strong until imp when being stuck on 9 range will start to hurt them. Still solid though.
8. Turks
Exactly the same as Chinese but also lack illumination as well which is the second most important Imperial Age monk technology. Like the Chinese, still solid until imp.
9. Mayans
No Redemption means they can't convert buildings and siege weapons. No illumination also means they are slow to reload. They can still get up to 12 range, still convert enemy monks and they have theocracy if needed.
10. Britons and Franks
No Redemption, no Atonement. They are the only civs outside the top 6 that have all 3 of the Imperial Age upgrades, so they're still very usable.
I won't go through any more but I'll just say that the Celts have the worst monks in the game, having none of the big 6 technologies available to them.
The Big 6 monk technologies at the monastery that actually matter i.e. they enhance the offensive capabilities of your monks.
Castle Age
Sanctity - +15 hit points, +20 for Aztecs (120 gold)
Redemption - Convert buildings and siege (475 gold)
Atonement - Convert monks (325 gold)
Imperial Age
Block Printing - Increase conversion range from 9 to 12 (200 gold)
Illumination - 50% faster reload (120 gold)
Theocracy - Only one monk reloads when performing group conversions (200 gold)
The order in which you research Sanctity, Redemption or Atonement isn't, in my opinion, that important and can vary.
In Imperial Age these upgrades should be researched in this order:
1. Block Printing is the most powerful Imperial Age upgrade. It's okay if you cannot reload your faith faster or the entire group of say 10 monks has to reload faith after a group conversion, but to be restricted to only 9 range is really dangerous at a time when when crossbows and skirmishers that get bracer will go up to 8 range, onagers have 8 range without siege engineers (9 if they do research it), bombard cannons have 12 range as standard, and melee units all get a little bit quicker in imp. Monks will need more time in order to convert things, and +3 range buys them that time.
2. Illumination should be researched second as it's 80 gold cheaper than theocracy and improves the performance of all your monks as individuals, which more important than their performance as a group.
3. Theocracy is a brilliant technology to research but unless you've already researched a minimum of block printing and illumination and you have at least 10 monks in play, then save your gold.
Monk Rankings
Tier 1
1. Aztecs
2. Teutons
3. Byzantines
4. Saracens
5. Spanish
6. Japanese
----
Tier 2
7. Chinese
8. Turks
----
Tier 3
9. Mayans
10. Britons and Franks
-----------------------
The top 8 civs are the only civs in the game with Redemption. They all have Atonement as well.
1. Aztecs
They have the all the technologies available, the +5 hit point bonus per technology researched, monks created 15% faster and the relic gold generation team bonus. They will probably end up on 75 hit points, because Faith and Heresy are too expensive, Herbal Medicine is useless and Fervor is bugged. In my experience the benefit you're most likely to get from the hit point bonus is being able to survive a direct mangonel shot. This alone is amazing. Large enough groups of Light Cavalry, Eagle Warriors and crossbows/arbalest/elite skirms will still make very short work of Aztec monks as even with more hit points they are still very slow and still have no armour of any kind.
2. Teutons
They have all the technologies available apart from Herbal Medicine (which is irrelevant). Their monks take 50% longer to be converted thanks to their team bonus. It's as if the entire civ has Faith researched from the start of the Dark Age. I think you could argue that because of this, they are the best civ in conversion wars. They also have a healing range of 8 instead of 4. That means healing range and conversion range are almost the equal. Now I understand how their team bonus works, I'd be more worried about using monks against them than I would against the Aztecs. Will we see the Teutons in MOA3? None of the settings for each game push people toward picking the Teutons.
3. Byzantines
They have all the technologies available apart from Herbal Medicine (which is irrelevant). Team bonus of 50% faster healing is good but not as good as the combination of the Teutons team bonus and healing range. Their fast imp capability may mean they are the first to get Block Printing, Illumination and Theocracy.
4. Saracens
They have the all the technologies available but have no monk related bonuses. Their Market bonus can help get you to Castle age faster and get the more monks out by using your market bonus to sell stone as seen in this game - https://www.youtube.com/watch?v=GJiZhfBg484
5. Spanish
They have the all the technologies available but have no monk related bonuses. There's the very slight potential advantage that their monasteries are built 30% faster so you could maybe grab the relics marginally quicker, but getting monks out a little earlier than other people who start building monasteries at the same time.
6. Japanese
They have all the technologies available apart from Heresy and they have no monk related bonuses. Heresy is rarely researched because it is so expensive. Even without Heresy, they are still a monk powerhouse.
7. Chinese
Exactly the same as the Japanese but they also lack Block Printing. Very strong until imp when being stuck on 9 range will start to hurt them. Still solid though.
8. Turks
Exactly the same as Chinese but also lack illumination as well which is the second most important Imperial Age monk technology. Like the Chinese, still solid until imp.
9. Mayans
No Redemption means they can't convert buildings and siege weapons. No illumination also means they are slow to reload. They can still get up to 12 range, still convert enemy monks and they have theocracy if needed.
10. Britons and Franks
No Redemption, no Atonement. They are the only civs outside the top 6 that have all 3 of the Imperial Age upgrades, so they're still very usable.
I won't go through any more but I'll just say that the Celts have the worst monks in the game, having none of the big 6 technologies available to them.