Combat Experience (Apocalypse)
In X-COM: Apocalypse, your Agents' stats can increase after combat. The improvements follow quite different rules than the system used in the first two games; there is no randomness, and there is no breakdown of stats into "primary" and "secondary".
Strength and psionic stats cannot be improved by combat. As such, any improvements in psionic stats must come from the Psi-Gym.
All combat improvements are subject to the ordinary stat caps (as displayed, these are 96 for Accuracy and 100 for everything else). XP is reset every mission.
Accuracy
Accuracy XP is gained every time a soldier hits an alien. Maximum Accuracy XP is 40.
Accuracy improves by ceiling((96-Accuracy)/100 * floor(Accuracy XP/4)). This is in terms of displayed Accuracy, not the number stored in the game files.
Stamina and TU
Stamina XP is gained when a soldier uses Stamina - i.e. by running.
Stamina improves by floor(Stamina XP/Stamina)/2 (second division is not rounded; Stamina can have half-points).
TU improves by min(floor(Stamina XP/Stamina), 5).
Reactions
Reactions XP is gained every time a soldier performs reaction fire. This does not appear to occur during Real-Time missions.
Reactions improves by min(floor(Reactions XP/Reactions), 5).
Health and Bravery
Health&Bravery XP is gained when a soldier kills aliens. The XP value is equal to the score value of all aliens killed by that soldier that mission.
Health improves by 1 if Health&Bravery XP is between 15 and 29, or by 2 if Health&Bravery XP is 30 or more.
Bravery improves by 10 if (Health&Bravery XP/Bravery) > 0.5 (e.g. Bravery would improve from 50 to 60 if a soldier killed 26 or more points worth of aliens in a mission).