Ok, so it is great to know one's way around the guitar and sure, hit and try is a good way to gain experience. But before we go any further, I urge you to give serious thought to getting all this stuff done by a professional.
Now, let's get this straight. All my info is derived from my tinkering around with my Ibanez GRG270 and a really crappy Gibson fake. So, even though I am trying to make this a general guitar article, you are bound to see some model-specific parts, which I just can't help.
My Ibanez GRG270 was made in China in 2002. It is a six-stringer with a Floyd Rose tremolo system, 22 frets, H-S-H pickup setup, basswood body, mother of pearl inlays and Enrnie Ball Extra slinky strings.
You need to keep the following stuff handy before you start:
Philips head screwdriver
An extensive L-Key set
Ruler (plastic preferably)
Dismantling And Cleaning
Push down your tremolo arm (tilt bridge towards headstock) and slide the ruler behind the bridge to provide support. This is basically to prevent the floating bridge from bending over with spring pressure and snapping your strings. Remove locking nuts. Remove all the strings. Flip guitar over and use the screwdriver to remove the backplate (the one containing the spring system, not the one housing the circuitry). In mine there are three springs. The springs all have one end attached to a screw plate (which can be adjusted) and the other hooks onto the bottom part of the bridge. Now remove the ruler and let the bridge sink in. Remove all the springs and you can detach the tremolo system completely. CAUTION - the screw plate (to which the springs are attached) is electrically grounded. Make sure you do not by any chance tear of the wire which is soldered on. If you do, keep a soldering kit handy. Now if all went well, you would be holding the Floyd Rose bridge in your hands. Dismantle it further by removing the sliding ends with the L-key these screw off from the tubes at the back side. (The tubes are where you insert your strings into) CAUTION - keep screws safely and separate. You will end up with loads of these pretty soon. It can't hurt to be meticulous. Now that you have this much out, start cleaning. Clean everything with the cotton rags. The sliders of the tremolo system may need some oiling if rust has accumulated. For the strings, I suggest this home remedy if they are really out of shape. Scrub down each with the sand paper lightly (careful not to take too much off, you might very easily go from an extra slinky gauge to extremely slinky!) then in a large utensil, fill some water and add a few drops of mild utensil cleansing liquid. Now put your strings in and boil for about 15 minutes. Cool and dry completely with a cotton cloth. Clean your frets (make sure the cloth is not wet) and the inlays. EVERYTHING!
If you were paying any attention, then reassembling should be no problem!
Setting Up And Tuning
Put the strings and use an electric tuner to get your Floyd Rose bridge parallel and in-tune. Actually the article in UG by Metal_Mad_Dog in the For-Beginners section is exhaustive enough, so I won't bother.
Now is the time to check your neck relief. Rule of the thumb - if you SEE it curved, you are in big trouble, otherwise faulty neck relief would lead to some fret buzz (around the middle, perhaps) now this involves adjusting the truss rod. Get the appropriate L-Key handy and make your way to the headstock. I there was fret buss around the middle, then most probably your truss rod is wound too tight. Loosen it be turning the L-Key clockwise (if viewed from the bridge side) about a quarter turn. This should sort out the problem. Retune and re-check for buzz. Be careful not to adjust the truss rod too much.
If the initial problem was that you can actually see the whole neck bent, then I suggest you do go to a repair shop. Ideally the neck relief is perfect if you have 0.3 - 0.5 m clearance at the eight fret. But since you probably won't have the tools to measure that, it is sufficient to just eliminate the fret buzz and be happy. Also, the string action can be controlled by two screws on both ends of the bridge. Use them to set whatever level gives you most playing comfort.
Fret buzz, neck relief, parallel Floyd Rose bridge all dealt with, your almost all done. To check you intonation, play each of the strings open and then fretted at the 12th fret. Ideally both notes should be same. But you probably won be getting off that easy.
Scenario 1. if the fretted note is flat: Loosen the string first. With the L-key remove the screw on the slider (on the bridge). Move the slider towards the headstock. Relock the screw and retune the string. Check intonation again.
Scenario 2. if the fretted note is sharp: Loosen the string first. With the L-key remove the screw on the slider (on the bridge). Move the slider away from the headstock. Relock the screw and retune the string. Check intonation again.
Now that you are all done, set your fine tuners to about halfway, retune the guitar. Lock the strings. Check tuning again and use fine tuners to make any adjustments.
If you followed all the steps with some patience and diligence, then you should end up with a nice clean, shiny, perfectly in tune guitar and also a wealth of experience.