Does first strike damage go on the stack separately from regular combat damage?

Combat damage — whether dealt in the first strike combat damage step or the regular combat damage step — is never placed on the stack. It is assigned and then dealt simultaneously as a special action. Players cannot respond between the assignment and the dealing of damage within a single damage step (CR 510.2).

What does happen is that there are two separate combat damage steps when at least one attacking or blocking creature has first strike or double strike (CR 510.5). The first step resolves damage only for creatures with first strike or double strike, and the second step handles all remaining creatures (and double strikers again). After each step, players receive priority and may cast instants or activate abilities before the next step begins.

So the key distinction is: players get a chance to act between the two damage steps, but not during either one. A creature killed in the first strike damage step will not deal damage in the regular damage step, which is the practical benefit of first strike.

Example: Your 2/2 first striker attacks into your opponent's 2/2 vanilla creature. In the first strike damage step, your creature deals 2 damage, killing the blocker. Your opponent cannot tap the dying blocker for mana or cast any spell to save it before it dies — but after that step resolves and state-based actions are checked, both players get priority before the second combat damage step (where the vanilla creature, now dead, deals no damage).