Does first strike damage resolve before regular combat damage for creatures with both first strike and double strike?

Combat damage is divided into two steps when any creature has first strike or double strike: the first strike combat damage step (CR 510.1) and the regular combat damage step (CR 511.1). A creature with double strike deals damage in both steps (CR 702.4b).

So a creature with both first strike and double strike behaves exactly like a creature with double strike alone — it gets to deal its combat damage twice. The first instance of damage lands in the first strike step, before creatures without first or double strike deal any damage at all. The second instance lands in the normal combat damage step alongside those other creatures.

This matters enormously for survivability. If the first-strike/double-strike creature deals lethal damage to a blocker in the first damage step, that blocker is removed from combat before the regular damage step, meaning it never gets to deal its damage back (CR 510.2).

Example: A 3/3 with double strike blocks a 4/4 with no special abilities. In the first strike damage step the 3/3 deals 3 damage to the 4/4. The 4/4 now has 3 damage marked on it but is still alive. In the regular combat damage step the 3/3 deals 3 more damage, totaling 6 — more than enough to destroy the 4/4 — while the 4/4 simultaneously deals its 4 damage to the 3/3, destroying it. Both creatures die, but the 3/3 dealt damage twice.

Key rules: CR 702.4b defines double strike as dealing both first-strike and regular combat damage; CR 510.1 governs the first strike combat damage step; CR 511.1 governs the regular combat damage step.