To obtain a new plane, you need to collect enough Blueprints (the progression of which can be checked in the plane’s menu, using each plane’s unique progression bar), and then unlock the plane, placing it on any free hangar spot.
Blueprints can be obtained from various rewards in-game and purchased from the in-game store. Please note that unless specified otherwise, Blueprints in all packs are generated randomly for all planes you have unlocked, but not yet obtained.
You can only unlock planes from available tiers. Requirements to unlock a higher tier are listed at the bottom of that tier’s table in the Planes progression tree.