With THREE hands-on videos (approx 20mins total) and pictures :D What more can you possibly ask for?
 
I'll cover the process of restringing in the first video and the process of setting up the balance after a tuning change or string gauge change as well as a few points on setting the intonation in the second vid. The third vid raises a point to blocking the bridge. So help yourselves!
 
You can hunt for additional videos on youtube. There are many out there, presenting different approaches. I just couldn't include all ideas in my vids because I'd need to make a couple more videos (maybe if I spoke faster I could include more...). So look around and combine the ideas that work best for you.
 
The basics
When you tighten strings you increase their pull on the bridge. Fixed bridges counter that tension by being solidly fixed on the wooden body. Tremolo systems, and especially the floating type (like Floyds), counter the tension with a set of springs mounted in a cavity at the back of the guitar.
So the Floyd bridge is basically hanging mid-air, stabilized by the pull of the strings on one side and of the springs on the other. The bridge's "zero point", the stability point, is where the tension of the springs is equal to that of the strings. Tuning up the strings increases their tension, as does sretching the springs.

Because of that fine and fragile balance between springs and strings, tuning such a guitar can be a very long and frustrating process. I've been there and now I'm gonna give you a few practical tips I learned along the way.
 
Tips on restringing

1) Don't remove all strings at the same time, if there is no pressing reason to do so (like taking off the pickguard, washing the fretboard, removing the bridge etc).
Remove one string at a time, put the new one and tune it up. You'll notice the bridge will tilt back when you remove the old string and all the other strings will go sharp and out of tune. That's ok. When you put the new one and tune it to the same pitch as the old one, the balance will be restored.
2) Keep the same gauge of strings, unless you really want to change gauge. Don't change gaugeall the time, just because you didn't care or didn't remember what gauge to use.
Strings of different gauge, produce different tension at the same pitch. Inevitably this will affect the fragile balance of the bridge. Tuning, intonation and zero point will all need to be readjusted. That's not something you want to do all the time. Trust me, it's a long boring process.
3) If you do remove all strings at once, there is a way of speeding up the tuning process when you put them back on.
With the new strings all in place, grab the arm/bar or whatever you want to call it and push it down until the bridge reaches an approximately horizontal position (the "zero point" position) and while keeping it there tune the strings up to pitch (your hand won't be steady so don't worry about perfection). This way you bypass the biggest part of the long process of consecutive tunings needed until the bridge reaches it's balance point. Release the bar and keep tuning the strings until balance is restored. It may take a few minutes, but it would have taken half an hour if you did it without the shortcut.
**) BLOCK THE BRIDGE:
Find a tool like a wrench, spoon or whatever you can get, that you can insert under the tail on the whammy and prevent it from sinking into the body when you loosen strings. The brisge must be blocked as close to its playing position as possible, otherwise it's pointless. In the following pic (taken from a re-wiring job, so don't freak out at the looks of it) I have used a wrench to block the bridge:

Blocking the bridge is very useful. It keeps it in place while you replace or loosen strings. This way, when you tune the string again, the bridge will be immediately balanced.
 
 
Setting the "zero point" position
The zero or balance point of the bridge must be identical to what is refered to as the horizontal or parallel position. With most floyd-type bridges this means literally that the bridge base plate must be parallel to the surface of the body. However, Ibanez Edge tremolos have a wedge profile design. Setting an Edge's top surface parallel with the body surface is very wrong, because that means the bridge is actually in a very pulled-back position. To determine the horizontal position of an Edge, locate the rectangle patch on the side of the bridge. Use the rectangle as your reference point and set it parallel to the the guitar top.
Setting the zero point position of a bridge is necessary after changing tuning or string gauge.
Tune the guitar well. Notice which way the bridge tilts. If it tilts forward, the strings are stronger than the springs. Tighten the screws (both the same amount) on the spring claw until the bridge has moved about halfway back towards horizontal position. The strings will have gone sharp. Loosen them back to pitch. Tune them well again. The bridge will have moved further back towards horizontality. Repeat as necessary. The closer to horizontal position you get, the smaller the adjustments you need to. If your bridge tilts backwards, then follow the same process but loosen the screws on the spring claw instead of tightening them and tinghten the strings instead of loosening them.
(see pics below)

 
This is a good place with loads of tips
on setup and maintense of floating bridges. It's focused on Ibanez models but physics are physics and know no brandnames.
http://www.ibanez-rules.com
 
Scroll down below the pics for more tips from readers of this blog.
 
This is how an Edge looks like well-balanced and unbalanced:
 
Parallel position:

Too loose (tilting forward):

And too tight (tilting backwards):

Notice how in this last one, the top surface of the trem base plate is parallel to the body. This is the proper position for Floyds and most copies, but it is a wrong position for an Edge.
 
These are the screws in the back of the guitar, that determine how tight your springs are:

 
 
 
Additional tips from readers
 
Comment by Tybalt On — 04-25-2007 05:40:36 EDT

 
 
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Hope this helped!
- RG -