The trick to staying on is to keep riding the ski while you're under. Sounds silly I know, but when you're learning you'll go under and think "yeah! Done it!" then you'll quit riding halfway through the move and bob back up, probably falling off the side in the process.
So, here's what you do. Once you're getting the nose under, build up enough speed that the ski will go down as far as the intake - that's pretty much when your head goes under. Once you're taking water up the nose, hit the throttle at about 3/4 open for a short blip - like a second at most. That will take you waaaaay down (make sure it's deep enough!).
OK, here's the crucial part. Right now you're pointing down and may have jinked a bit to the side when you went in. So, you lean back, and hit the throttle again for about the same type of burst. That will level the ski out and the acceleration will knock your body back into the right position for the exit. At this point, don't forget that you still have steering! If you're teetering to the side, then do what you would do if you were still breathing air: twist the bars when you blip the throttle to regain your balance.
The final part is easy: one more blip on the throttle brings you triumphantly to the surface with aplomb, rather than bobbing back up like you didn't mean to do it :arms:
Like I said: just because you're under doesn't mean that you should stop riding the ski. Getting under is only half a sub....
Good luck!
(Oh, one last thing: you don't always come up from a sub in the same direction that you went in, so make sure you have space either side of you when you're going deep. A fun variation is to do a 180 underwater on purpose, and come up traveling in the opposite direction....)