I mount the plane on a piece of acrylic (plexiglas) rod, diameter depends on scale. I use 3/8 for 1:32 for 1:32 scale and bombers, 1:4 for 1:48, and 1:8 or even 3:32 for really small, light stuff. You need some reinforcement iside- I use wood. You need to get hole drilled and reinforcement before glueing fuselage together if the sides are split vertically. Get a good fit an then glue the rod in when model complete.
For a base I use a piece of MDF, either half or three/quarter, with wood molding on edges. Then I go to Google Earth and go to an area that is relevant to the operations of the bird- airbase or operations. For instance, I am making a diorama (1:144) of two F-4s escorting a TU-95 over the Bering Straits. Found a good view of an island, which I will print out when the base is cut to size, and glue to top with contact cement.
While building, if need to, I make a working base from scrap particle board or plywood with a simple hole in middle. If you need to do any painting after inserting rod, just cover the rod with masking tape.