In a nutshell, X-wing is played over a series of turns.
You will begin each turn by secretly assigning a maneuver to each of your ships.
Once all ships have been assigned a maneuver for the turn, those maneuvers are resolved one by one.
The order in which maneuvers are resolved is determined on a ship-by-ship basis, according to Pilot Skill.
Less-skilled pilots such as the humble Academy TIE Fighter pilot perform their maneuvers early in the turn.
Pilots of greater skill such as Han Solo, Luke Skywalker, or Darth Vader perform their maneuvers later in the turn.
Immediately after a ship has maneuvered, it may perform an action. These actions take many forms, but generally provide some additional maneuvering capacity or some offensive/defensive benefits.
Once all ships have maneuvered and taken actions, the combat phase begins and ships exchange fire!
The order in which ships shoot is determined by Pilot Skill, with more skilled pilots shooting before less-skilled pilots.
For more information, I suggest watching one of the following:
For starters you'll need an X-wing Miniatures Game Core Set, or an X-wing Miniatures Game: The Force Awakens Core Set.
These each contain one Rebel starfighter, two Imperial starfighters, some pilot & upgrade options for each of those starfighters, a set of maneuver templates, a damage deck, tokens, dice, and the various other things you'll need for your first game or two. While not strictly necessary for your first game, it is a good idea for each player to have their own Core.
Chances are excellent
that you'll want to expand your fleet almost immediately.
While X-wing is highly adaptable to various squadron point limits, standard dogfights give you 100 squadron points to build with - and you'll have to purchase additional ships to reach this.
Popular bang-for-your-buck choices include the Rebel Aces expansion and the TIE Interceptor expansion - but
you can pick up anything that looks cool.
Which brings us to...