Here is my How to use the Cardinals effectively guide , provided by my experience. 

First of all how to be able to get to the Cardinals them self?

You need to reach a Palace level 21. Than you are able to build the Conclave Of Cardinals.Than you will have access to them. Every time you upgrade that building you will get access to new Cardinals and their levels.

Second as you may know in order your army to be strong you will need just one troop type to play with. That is because several reasons:

• Your hero gear benefits only 1 to 2 troop type troops.(unless you have goodly Arsenal hero gear that gives bonuses to all of your army, but that is usually when you are like 10 000 000 000 in power.
• Your Inquisitor gear benefits only 1 to 2 troop type troops.(unless you have goodly Inquisitor gear that gives bonuses to all of your army, but that is usually when you are like 10 000 000 000 in power.
  
• The Cardinals also support 1 or 2 troop too.
  
• Upgrades from Alchemical Lab are so expensive that you cannot afford to upgrade more than one troop type.
  

Next to know about Cardinals is that for upgrading each cardinal requires different kind of resource.(Resources, Gold or Trophy Of Arms)

If you wonder what is Trophy Of Arms and how to get it, that are trophy witch you get when you kill troops from different Kingdom. For 1k troops (no mater of their Tier) you get 1 Trophy Of Arms. 

That is extremely low and upgrading a Cardinal with them is almost impossible. So my tip is use Gold or Resources!!!

When you level up a cardinal you will be able to summon him. After you summon him it last for One hour ! Then you have to wait the Cardinal to recover witch is around 2 days (The higher Cardinal level is the shorter the recover time will become, but not shorter than 2 days)

As The Cardinal leveling up is expensive you need to pick a troop type first. Then see who of the Cardinals benefit your troop type and will be easy to level up. I choosed the Spear cardinal witch is The Light Bringer, because he is upgraded by resources and it`s the easiest to level.

Important thing to know is that when you are Upgrading using ress the Cardinal upgrade will need time to process! But when you are upgrading using Gold and Trophy Of Arms for example the upgrade is instant 

Very important ting to know is that Cardinals has 3 types of boost:

1. Temporary (when you summon the Cardinal). That last only in that One hour time period.The temporary bonuses are 3 (for example Spearmen Offence, Scout Defence and Marching Speed)
   
2. Permanent. One of the 4 bonuses that each Cardinal gives is permanent it benefits your troops non stop.
   
3. Bonus against each other. (That is bonus which depends on from which chamber is the Cardinal. Information about Chembers and conflicts you will see down below)

The maximum level of all Cardinals is lvl 10

The maximum % per boost is 50% 

(as the most of the studies and hero skills goes the bigger the upgrading lvl is the bigger % it gives per one point)

See more about stats and upgrading here : https://plarium.com/forum/en/throne-kingdom-at-war/599_fan-guides/122194_cardinal-info-and-stats/

Interesting ting bout Cardinals is that they give huge power points. (for example my Light Bringer is lvl 10 and it made my power + 745 Million in power.) That is more like a minus because it looks like you are very powerful but 745 m of your power is just your Cardinal)

The last ting that you have to know about them is that there are 3 cardinals Chambers that are in conflict.

Twilight Chamber (Hugo The Darkest, Helene The Midnight)

The Sun Chamber (Giorno The Lightbringer, Edmondo The Ciymson)

The Moon Chamber (Heidar The Shining, Oddgeir The Crescent, Laurie Argent)

These 3 chambers are in conflict. Because of that Cardinals has bonuses against each other.

Finally my strategy of using the Cardinals.

First pick your main Cardinal. That will be the one that is giving Permanent Health bonus to the troop type that you use. For example i use Spear type troops and i picked the Giorno Light Bringer.

Upgrade him as far as you can! Doing that you already use his Temporary bonus to your Spears health.

Knowing that by summoning him you will have just one hour period to get his Temporary bonuses, you have to use him effectively !!!

.What that means?I will tell you what i am doing to be effective when i summon my strongest Cardinal.

   First i take notebook and a pen . Then i search for enemies and write their coordinates down on the paper (you can also use land marks)

 When i find several enemies to fight with i activate my hero fighting profile, my military boost (Army off, Army def, reduce enemy- health, offence and deffence) my Obscuration boost so the enemy will not be able to see you hero, inq and cardinal information and then i summon my cardinal ! Doing that i know that i am fully buffed and i have already found all the enemies i want to hit and i will not lose time searching for them. Fast way to access your cardinals if you are playing from PC (computer) is with the "V"  letter on your keyboard.

That is the strategy of using your primary Cardinal. But there will be other Cardinals that you will be able to level up at least to level one. So when your primary Cardinal is recovering you can also use the other ones. The boosts they give are not only military ones. For example 

Hugo The Darkest gives bonus to training troops, Edmondo The Crimson gives bonus to buildng etc. So even that Cardinals are lvl 1 or lvl 2 and that bonuses are 1% or 2 % , we all now that with time Studies and Building time is taking mounts so every bonus is important !

And you can also use the conflicts between chambers too. If your primary Cardinal is recovering if you have information on the enemy Cardinal you can summon one that has bonus against the enemy one! Or when you see that your military power is equal to the enemies one and you wonder to fight or not it`s pretty possible exactly that Cardinal conflict bonus to decide the fight! 

 For example you and your opponent, bought with Spear class troops , with equal gear  and  equal power. There are 2 cardinals that benefits your spear type troops. 

1. Giorno the Lightbringer , that gives Spear men defence bonus (temporary) and Spear men health bonus(permanent) - Sun Chamber
   
2. Oddger the Crescent , that gives Spear men offence bonus (temporary) and army speed bonus (temporary) - Moon Chamber

Let`s say you and your opponent bought have that 2 Cardinals maxed up. And you have them bought ready to summon but the opponent has been using Girno the Lightbringer shortly and he has just Oddger the Crescent.So he is summoning him, also strong cardinal he got that health bonus from Girno the Lightbringer, eaven that he is restoring , because that bonus is permanent) Now you have to decide witch cardinal to use.And knowing about the cardinal chamber conflict  you go to that page in the game !!! 

and when you see that the enemy cardinal is Oddger the Crescent  member of     The Moon Chamber     if you choose to summon Giorno the Lightbringer member of         The Sun Chamber       i will have that sweet 4 % Army Offence bonus and will be easy to defeat your enemy!!!

I hope i have been in help to someone, because i know how i was struggling to find how Cardinals works.