Is there any way within the game to replay missions, ala the NFS games? I figured there is a way to do it with an independent program but I've neither the patience nor the brains for that, plus it's not really for show but for me. When I'm flying I'm so damn busy concentrating on not screwing up I never get to sit back and just watch, which I'd like to do.

No such option within the game as far as I know.

So make the most of the moments where you can actually relax : orbits, post-touchdown, pre-launch, waits to get to the position (with or without node) of your next thrust. Heck, even during liftoff, there's a little moment (when your apoapsis is right and you've cut your engines and are just waiting for your horizontal burn moment) where you can untangle your toes a bit.

But, generally, relaxed moments are the reward of the tense ones. But as your missions grow in ambition and scale, the idle times will take more and more room. Enjoy the time between a mission's "steps". The time around Kerbin before pushing towards the Mun, the time in transit before braking, the time orbiting the Mun before dropping, the time standing on the Mun before taking off again, etc...

Apart from that, you can indeed make video captures through, well, these separate video tools. Even if it's just for yourself. But, I'd say, re-watching AVI video files recording or your game wouldn't be the same as an in-game carmageddon-like replay...