School Of Monsterology

Despite the size of the Gilded Coast, the majority of their monsters are contained in one city: Atropolis. When the Iris Of Hell opened nearby, the martially skilled charged on, only to paint the streets in their own blood. The scholars were more patient, barricading themselves in the Cathedral Of Feyte. There they would study the monsters overrunning their city. Their research provided soldiers with weaknesses to exploit and turn the tide against the never ending flood of Hellspawn.

While the aristocrats of Clothopolis looked down on their sister city for the demonic infestation they purposefully kept alive, the students of Lydia’s School Of Monsters prided themselves on the very few fiends that escaped. After all, to study monsters, you require live subjects. 
~ The Tale Of The First Monsterology Wizards

Field Research

Beginning when you select this school at 2nd level, you have a requirement to research in the wilds, to study monsters in the area. Due to this, you may need to remain stealthy and quick! Your walking speed increases by 5ft, and you ignore nonmagical difficult terrain. Additionally; while Lightly Obscured, if a creature isn't aware of your presence, you are treated as Invisible.

Volo's Guide

At 2nd level, you gain proficiency in the Arcana skill, and double your Proficiency Bonus to checks made with it. Additionally, your spellbook becomes a Book Of Monsters, letting you consult it for information on monstrous foes. When you roll initiative, before combat begins, you may expend your reaction to study a creature you can see and hear, searching for their information in your Book Of Monsters. When you do so, You learn the targets species name, resistances, immunities, vulnerabilities, and most defining ability and action. You may target an additional creature in this way when you reach the 6th (2 creatures), 10th (3 creatures), and 17th levels (4 creatures). You must have access to your Book Of Monsters to use this feature.

Tasha's Exploitation

Starting at 6th level, you research into more unique cantrips to better counteract more unique foes. You learn up to two Cantrips from any spell list, but each must deal damage. When a creature takes damage it is vulnerable to from a cantrip you cast, for the first time in combat, you may force that creature to make a Wisdom saving throw against your Spell Save Dc, becoming Frightened of you until the end of your next turn, on a failed save.

Additionally, can spend a 1 minute ritual inspecting an incapacitated creature. If you complete this ritual, you add it’s Original Species to your Noted List. When you make a spell attack roll against a creature from your Noted List, you may add your Intelligence modifier to the attacks damage roll. You may use your Volo's Guide feature on a creature who's species is in your Noted List, at any time, as an action.

Mordenkainens Eye For Weakness

At 10th level, you become able to notice the vulnerabilities of creatures even when others may never notice. As an action, you can target a creature within 30ft of you that you can see, and make an Arcana check, the Dc being equal to the creatures Level/Cr. On a successful check, you may choose a damage type the creature is not already resistant or immune to, making it vulnerable to that type of damage, when you deal that type of damage to it. Many may not notice, but you can tell the Vrock's underbelly is far more weak to Thunder damage than most realize.

Page By Page

Starting at 14th level, when you study a creature with your Volo's Guide feature, you instead have access to the stat block of the creature's original species. You learn the Stat Block's of all creatures in your Noted List.
At level 20 you have access to the Monster Manual at any time because it's level 20, nobody will complain.