What is the recommended procedure for adjusting the fit of my bike? I have a road bike which is a reasonable fit, what is the recommended procedure for adjusting the seat position and handlebar height to get the "perfect fit"?

A complete answer should include what order to make adjustments; how to determine height of saddle, fore and aft position of saddle on rails and height of handlebars. Also what testing is required. Is this all done on feel or are there basic measurements and mecahnics that affect the adjustments? Does it help to have a turbo-trainer and large mirror or video camera?

Do the same principles apply for hybrid and/or mountain bikes? What changes to the process are needed for other bike types?

I know that this is a service you can pay for at some pro-cycling shops, but since I have a number of different bikes, I would like to be able to get a good fit for each of them at home for a minimal layout. And also, by understanding the process and doing it myself, getting a better understanding between me and the bike...
There's quite a lot of discussion on the subject of fit, and some folks have considerably different ideas in this regard.
I use the old-school methods to get close, then try to "dial in" things according to what my body tells me.

Seat height should be such that you can easily pedal "circles" without rocking or stretching, nor by over-flexing the knee.  It's hard to describe...Essentially the leg should not be fully extended at the bottom of the stroke.  Just short of being fully extended.
The thigh should be more-or-less level at the top of the stroke.

The fore-and-aft measurement is often neglected or even ignored.   Again, the classic measurement is that with the pedals level (3 and 9 o'clock) a plumb line dropped from the bony process just below your kneecap should intersect the pedal's shaft.
That's assuming you're sitting on the saddle properly, with your "sit bones" on the heavily-padded portion of the saddle.
Saddle attitude is one of personal preference; I prefer just a few points nose-down from dead level.
"Reach" is somewhat dependent on the type of bike.   On roadsters, you want the reach to the hoods such that you can ride with a relaxed upper body at a reasonable angle.  Some authorities advise such-and-such a percentage of weight on the hands, and such-and-such on the seat.  You don't want to be totally supporting your upper body with your arms.
I've seen folks who maintain that these things are "all wrong", and propose all sorts of ideas for getting the perfect position.    I know I can ride comfortably and without hurting anything using these methods....