Well... first of all, After Burner is a full game. But to make it more of a console-style game that still has an arcade feel...
Start with a large world map with many distinct regions and many bases to begin from (including the Sega Enterprise air craft carrier, of course, but many others too). The regions are not only geographical features but also correspond to altitude, so in any given region there are events that could occur depending on whether you're flying low to the ground, above the clouds, or in between.
The player accepts a mission objective from a list of several to choose from at any time. Each mission has several secondary objectives and one main goal. The objective is usually several regions away on the map, so the player has some choices about how to get there and possibly some incentive, like temporary power-ups as well as long-term XP) that can be gained depending on which regions they go through.
Within each region are challenges that are similar to what we're used to in After Burner -- enemies flying in formations, with the goal of maximizing score and staying alive. The big difference is that you're not on rails -- you can turn in any direction. Still, you'll max out your score by taking out an entire formation in one go rather than piece by piece. Bosses would include giant flying fortresses and bases that must be attacked from every direction.
In this After Burner, you've got to manage your resources a bit more closely and do your own refueling and reloading of weapons. This could be done in a streamlined way to keep the fast pace, with a minigame to maximize what you gain from the refueling/reloading plane or base. There would also be a long-term leveling system, gradually enabling you to unlock and install upgrades and weapons loadouts for your planes. As is usually included in my ideas for these games, there would be a livery editor for your planes and players would also have different types of hangars to choose from and decorate to keep their planes. The hangar would naturally have the original arcade After Burner inside it.
Although the primary focus is on fighters, there could be additional types of aircraft for specialized mission objectives... bombers, recon planes, and helicopters for rescue missions.
The game's single player campaign would be set up similarly to an RPG, with a main storyline that can be advanced at the player's discretion by going to specific places and fulfilling certain requirements to trigger story elements. But there would also be the sandbox element of being able to just fly around for fun, take on side missions, and pull off aerobatic tricks for points and XP.
The multiplayer aspect would be integrated so that you could see other players flying around in your world and you could accept or ignore challenges from them for dogfights, races or other skill challenges. There would also be co-op missions with objectives and enemies that level up with the players. Finally, there could be separate lobbies for more elaborate team multiplayer modes including the usual suspects -- team dogfights, capture the flag, escort missions and so on.
