The first aircraft unit, the Douglas C-47 transport, becomes available after the fourth mission of Operation 1. It costs 20,000 prestige points to unlock it.
To recruit the C-47, select the fifth mission then go to the Recruit Unit menu from the Third Army Units tab.
Personally though, I would save your prestige points until later during Operation 2 and spend them on the P-51D Mustang. The ground attack capability it gives is more useful than the C-47 for completing missions and it costs 5,000 fewer prestige points.
If you can't wait to pop the achievement, recruit the C47 during mission 5 then exit rather than start the mission. You then need to either Load Mission from the main screen (rather than Resume Campaign, as it goes back to the Unit screen you just left) or quit to dashboard, reload and then Resume Campaign. That way, you can get it early but keep your prestige points for something more useful than a transport aircraft.