Does first strike damage resolve on the stack before regular combat damage?

First strike (and double strike) damage does not go on the stack at all — no combat damage ever uses the stack. Instead, when at least one attacking or blocking creature has first strike or double strike, the game creates an additional first strike combat damage step before the normal combat damage step (CR 510.1).

During the first strike combat damage step, only creatures with first strike or double strike assign and deal their combat damage. Then players receive priority and may cast spells or activate abilities. After that phase ends, the game moves to the regular combat damage step, where creatures without first strike (and double strikers again) deal their damage (CR 510.2).

Because combat damage is never placed on the stack (CR 510.1), it cannot be responded to the way an instant can. What players can do is act during the priority window between the two damage steps — for example, removing a blocker after first strike damage has already killed it, or saving a creature that took first strike damage before it would die in the regular step.

Example: Your 2/2 first striker attacks and is blocked by your opponent's 3/3 vanilla creature. In the first strike damage step, your 2/2 deals 2 damage to the 3/3. Your opponent cannot respond to that damage, but they can cast an instant like Giant Growth during the priority window afterward. If they do nothing, the game moves to the regular step where the 3/3 (now at 2 damage) deals 3 damage to your 2/2, trading — but if the 3/3 had only 2 toughness, it would die before ever dealing damage.

In summary: the two damage steps are separate game events with a priority window between them, but neither step involves the stack (CR 510.1, 510.2).