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This is my first time playing a Paladin. He just reached Level 3 and I am torn between all the Sacred Oaths. They all seem to have their advantages and disadvantages. I use the Great Weapon fighting style, but have no feats.
What I would like to know, is which one does the most damage of any kind?
None of the Oaths really do more damage than the others as such. The only Oath feature that directly impacts your damage is the Oath of Devotion's level 20 ability, Holy Nimbus - it deals 10 radiant damage to any enemies within 60 feet of you. So, to answer your question directly, Oath of Devotion does the most damage.
This is, obviously, a fairly ingenuous answer. The real answer is that you can't look purely for damage.
For example, both the Devotion and Vengeance Oaths have features that increase the accuracy of your attacks, which in the long-term, will increase your damage.
On the other hand, the Oath of the Ancients has an ability that makes you and your allies much tankier and harder to kill. How does that boost damage? Well, you can't deal damage if you're dead, so anything that keeps you alive is also indirectly boosting your damage.
Then there's things like spells. The Oath of Vengeance gives you access to hunter's mark, which potentially lets you add 1d6 damage to every attack you make. But in practice, it's not quite so simple - it requires concentration, and you also have to move it around to make sure it's on the enemy you're attacking. And, obviously, it requires you to make a lot of attacks to maximize the damage it gives.
As a counterexample, the Oath of the Crown gives access to spirit guardians, which deals damage to enemies who come near you - under the right circumstances, this will vastly outdamage hunter's mark. But in the wrong circumstances, it might be completely useless.
So, in the end, there's no easy answer - you'll have to look at all the Oaths and determine which one you think will work best for you.
As stated, there is no easy answer to this. For example, a Wizard can make a hard encounter an easy one without throwing a single fireball. That is much more impactful that just raw damage.
BUT
If what you want is to feel powerful in terms of doing damage, I would say that the Oath of Vengeance paired with Polearm Master feat and Hunter's Mark is a good start for a melee oriented Paladin. At level 5 you will have 2 attacks first round to a new enemy with an 1d6 extra per attack that hits (your bonus action is used to cast/re-position Hunter's Mark), and 3 attacks afterwards. And, as a bonus on though fights, you have 3 attacks to apply Divine Smite and Improve Divine Smite. Even if you lose the concentration of Hunter's Mark, you still have the bonus attacks.
Pole Mastery
When you take the Attack action and attack with only a glaive, halberd, or quarterstaff, you can use a bonus action to make a melee attack with the opposite end of the weapon. The weapon’s damage die for this attack is a d4, and the attack deals bludgeoning damage.
For a caster-ish Paladin, Oath of the Ancient is also very good. A well placed Moon Beam can control the battlefield and/or decimate enemies. And, if you have a good charisma or plan to have it (and don't want to multi-class), take Magic Initiate Warlock feat with Hex as the level 1 spell and Eldritch Blast as one of the cantrips. This option gives you a great range attack option, something that the paladin kind of lacks, and Hex for though fights.
