Depends on their level. Based on what you said, they are either level 3 or 4. They can prepare half their paladin level rounded down + charisma modifier.
At level 3, they can prepare 2 additional spells, and level 4 can prepare 3 additional spells. If they are level 3 and when they level up to 4 they use their ASI to increase charisma to a +2 modifier, then they could prepare 4 spells.
I have a PC who has a paladin
Charisma Modifier is +1
They have oath of Vengeance
On their character sheet generated here, they have 2 spells that state "always prepared"
So how many additional spells can they prepare today
Depends on their level. Based on what you said, they are either level 3 or 4. They can prepare half their paladin level rounded down + charisma modifier.
At level 3, they can prepare 2 additional spells, and level 4 can prepare 3 additional spells. If they are level 3 and when they level up to 4 they use their ASI to increase charisma to a +2 modifier, then they could prepare 4 spells.
So at level 3
4 spells(2 which are always prepared & 2 of their own chosen)
Correct. *For that level and Charisma.
Thanks for clarification