I've never really messed with or tweaked on the bridge of my Gibson Explorer before, and I feel that the intonation is in definite need of tweaking now that I increased the gauge of the strings recently. I've already adjusted the truss rod properly, but I need to know how to intonate it.

I'm assuming you hit the open string and then hold it at 12th fret and keep tighting/loosening each screw until the tone matches from each fret, amirite?
Basicaly, the natural harmonic on the 12th fret of a string has to be EXACTLY the same note (though an octave above) as the same string played open.

And do that for each string
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tune the open strings to pitch.
play a 12 fret natural harmonic and play the 12 fret fretted down. if the fretted note is sharper, you have to lengthen the strings. if it's flatter, then you gotta shorten it. everytime you adjust the saddles and then check, that string will be out of tune, so just tune it back up and repeat on all the strings until it's perfect
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