Here's a quick and dirty "guide" to combat:
Your five bridge officers are the only active participants in a battle, regardless of whether it's ship or ground combat, so if you have stronger officers in the rear stations, you'll want to bring those to the bridge and throw the weaker ones in the back.
Initiative determines how many turns it takes before a crewmember can act. The higher the character's initiative is, the fewer turns of downtime they'll have before acting.
Clicking the icon of the selected action will bring up a menu with up to four options for actions. The game will autoselect actions, but you can override the selection as needed.
When it's your side's turn to act, you'll be presented with "cards" based on the traits of your officers. Those cards will boost the effects of a selected crewmember's action, based on the stat(s) of the selected crewmember. Each card used takes a turn. If you run out of playable cards (or don't like the cards that remain) you can burn a turn to draw an additional card.
Your crew will target enemies at random during ground fights. A unit that has its HP dropped to 0 will be unable to act for the remainder of the fight.
Once all of the crew of one side has 0 HP, the fight ends.