The first thing you need is an empty Hangar (of any kind). Tap it to open the plane-buying tab in the Store. Once there, select the plane you need from those available at your level and press Buy. Keep in mind that only one plane can be kept in one hangar.
Open the Warehouse (box icon on the lower toolbar) and go to the tab with planes. You'll see the statuses of the planes next to them (in flight, broken, etc.). You can only sell working planes that are in the hangars. Once you've settled on a plane, select it and press Sell.