Does first strike damage kill a creature before it can deal regular combat damage?

Combat damage is divided into two steps when any creature has first strike or double strike (CR 510.1). In the first combat damage step, only creatures with first strike or double strike assign and deal their damage. State-based actions are then checked — meaning creatures with lethal damage are destroyed before the second step even begins (CR 704.5g).

In the second combat damage step, creatures without first strike (and double strikers again) deal their damage. If a normal creature was already killed by first strike damage, it is no longer on the battlefield and deals no damage in step two (CR 510.2).

This only applies when at least one creature in combat has first strike or double strike. If no creatures have either ability, there is only a single combat damage step and both creatures damage each other simultaneously (CR 510.1).

Example: Your 2/2 with first strike blocks your opponent's 4/2 vanilla creature. In the first damage step, your 2/2 deals 2 damage to the 4/2, which is lethal — the 4/2 is destroyed by state-based actions. In the second damage step, the 4/2 is gone and deals no damage, so your 2/2 survives unharmed.