I've completed the mission 3 times now, and it still won't let me move onto the next one. I'm not sure if this is just an iOS bug, since I've only played it on the ipad so far.

That's a repeatable mission. It's there so you can take it again in case you forget.

Oh, well it's says there's another mission available after it. I don't see it.

Once you finish the Basic Flight Test, the game becomes much more open. You're not required to take this mission or that mission in order to "progress"

Try some of the other ones. Especially the Courier missions for easy trade XP and some good starting cash.