Once you know what you're doing, it's not all too hard to understand (now, I might present this differently than more experienced persons on the forum, but it's still a way to do it).
You'd need to start out with several SEXPs: one that locks player controls so they can't move out of the way and mess up the sequence (visually, at least), one to create-warp-effect, one to flash the jump out HUD gauge, and one to play the warp-out sound effect. All of those will fire before the jump officially happens, and timing is going to be important.
When you've got that part down, you're probably going to need a very brief fade-out to white SEXP (I'm not totally sure about that. The warp effect might block vision enough to pull it off without it), and a change-coordinates SEXP. Be sure to set-facing the direction you want it to be. Unlock player controls, and the player has 'jumped.'
For AI ships, you can do the 'jump' one of two ways. Way one takes the existing ships and uses coordinate manipulation to bring them to roughly the same area as the player (be sure to use create-warp-effect with relative coordinates to the ship, or it'll look like they just disappeared and/or reappeared). As a precaution, I'd clear orders to make sure they don't end up chasing after a ship that is now several hundred or thousand kilometers away.
Way two is a little more simple from a jumping perspective, but more complex from a units standpoint. Instead of using coordinate manipulation, just have the first ship jump out, and have a ship with hidden characters in its name jump in near the player. In order to make damage carry over, use the copy-damage SEXP Battuta was talking about.
NOTE: Understand that this is from someone who has never actually FREDed a player jump in mission.
