Does first strike damage go on the stack or is it simply dealt before regular combat damage?

Combat damage — including first strike damage — never goes on the stack. It is assigned and dealt simultaneously within its respective combat damage step, but it does not use the stack at all. This applies to both first strike and regular combat damage (CR 509.2).

What makes first strike (and double strike) special is that the game creates an additional combat damage step just for creatures with first strike or double strike (CR 509.5). This extra step happens before the normal combat damage step. Creatures that only have regular combat damage deal their damage in the later step, so they may already be dead before they can strike back.

During each combat damage step, the active player assigns damage from their creatures, then the non-active player does the same — all simultaneously — and then all that damage is dealt at once without any player receiving priority in between (CR 509.2, 509.3). Since nothing goes on the stack, players cannot respond to the damage assignment itself with instants or abilities before the damage is dealt within that same step.

Concrete example: Your 2/2 with first strike attacks into your opponent's 2/2 with no abilities. In the first strike combat damage step, your creature deals 2 damage to the 2/2, destroying it. When the regular combat damage step occurs, the opponent's 2/2 is already in the graveyard and deals no damage — your creature survives unharmed.

Players do receive priority after each combat damage step ends, so after first strike damage is dealt and creatures with lethal damage are moved to the graveyard via state-based actions, players may cast instants or activate abilities before the regular combat damage step begins (CR 509.4, 117.3b).