I think if we are to look at the likely technological trajectory of commercial or militarized spacecraft (i.e., not the limited scientific missions that we are currently seeing), initial craft will almost certainly be small, relatively short range craft which will look similar in many ways to current aircraft. I would expect that the first few generations will be piloted by fighter pilots or test pilots.
By the time we get to a sci-fi level of large spacefaring ships which can spend extended lengths of time in space with a large crew, the organizations which created these craft will have developed their own specialized training pipeline. Although at that point I would argue that 3-Dimensionality aside, the "Bridge" or control centre will probably have more in common with a ship's Bridge or ship/submarine Ops Room/control room than a cockpit (in the sense that you probably require more than a cockpit-sized control area for multiple people. Maybe an AWACS back end?).
Once we get really far into the future and into interstellar travel, the mathematical complexity of almost anything we do will probably see a very smart AI doing all the heavy lifting while us squishy humans are either in some sort of stasis or at least just monitoring diagnostics to make sure that the very smart AI is doing what we expect it to. Maybe it's actually network admins who will be the Captain Kirks of the future, although that doesn't make for very exciting movies.