Damage is calculated as base damage times damage bonus times crit multiplier times defense multiplier times res multiplier times vulnerability times broken/toughness multiplier.
Most of these are similar to their counterparts in Genshin. The new multipliers are the last two: vulnerability is a multiplicative damage increase. Characters like Jing Yuan, Himeko, etc. have skills that can increase damage taken/increase enemy vulnerability.
The toughness multiplier is related to whether their toughness gauge (armor) is broken or not. If it is not broken, all damage is multiplied by 0.9. If it is broken, it is multiplied by 1.
Some other notes: the defense multiplier is calculated in a similar way as that in Genshin. The basic result is that if the character is lv90 and enemy is lv90, then the def multiplier is around 1/2 (so it halves the damage). The enemy's defense scales with their level, as in Genshin.
Resistance is calculated a bit differently in star rail, compared to Genshin. The formula is a bit simplier. The res multiplier is just 1 minus the res of the target plus the res penetration of the attack. This 'res penetration' is akin to res shred / res ignoring. But it should be noted that the HSR formula is linear. There is no nonlinear tapering relationship for res >= 100%. In fact if an enemy has over 100% res that could reduce the multiplier to 0. So that keeps HSR elemental res within a more bounded range. Typically, enemies will have 0, 20, or 40% res to different elements.
When the enemy's toughness gauge (armor) is depleted using the appropriate elements, they will enter into a weakness break phase, where they are unable to attack, they take increased damage, and they take break damage / dot. The type of break damage they take (break effect) depends on the element used to break their armor, the character's level, and the enemy's max toughness.
So similar to transformative reactions in Genshin, the break damage depends on character level. However, the break damage also factors in def, res, vulnerability, and the broken multiplier.
Different elements have different break effects, debuffs, and dot.
Weakness break gives some more uniqueness to each element, but does not change that the elements do not really interact / react with each other. A simple idea for incorporating reactions is to look at the TCG card game in Genshin. For example, if pyro is applied to a card, and then anemo is applied, it will create a swirl reaction, dealing damage to surrounding enemies. Similarly, in star rail, one can implement a similar kind of 'reaction': if one applies fire to an enemy, the fire status will stick on the enemy (for say 2 turns), if wind is then applied, it will swirl, remove the fire status, and deal fire damage to adjacent targets (and deplete fire weakness on the shield if any).
Similarly, lightning swirl and ice swirl can be implemented. Moreover, the overloaded reaction can also be transferred: lightning + fire deals aoe fire damage. Melt can also be transferred: fire + ice deals increased / amplified damage. Superconduct could possibly work: ice + lightning deals aoe ice damage and shreds physical res of target. Physical need not react with other elements. The only remaining elements are quantum and imaginary, so it would be up to the game architects on how these two non-Genshin elements react with other elements, if at all.
The flip side to this is that enemies can also deal reaction damage on the player.
Because of Silver Wolf, mono teams can be the norm in HSR. Resonances can give more uniqueness to the elements. Much of the following is inherited from Genshin: