I'm trying to intonate an Ibanez GIO GRG120EX to drop A#, but when I get the 12 fret in tune the open string it's out of tune and vice-versa. So far I've moved my pickups so they are not too close to the strings and put new strings. How do I set my guitar intonation?
While you're setting the intonation, don't get too fussy about the actual tuning until you're just about there.
Think about it producing the same pitch at the 12th fret as the open string or harmonic, rather than the correct pitch.
It sounds like you're expecting the intonation to magically change one pitch without affecting the other. Each move of the bridge saddle will affect both the open pitch & 12th fret pitch - because it will change both the string length & the string tension.
Depending on how far you need to move the saddle, this may take it quite a way from correct pitch each time you adjust, so correct it a bit after each move, but don't get hung up on it being absolutely perfect while you're adjusting it, just set it afterwards.
As you may well have to slacken the string each time to make the bridge move smoothly there can be a fair amount of guesswork involved in the first few moves, as Sparqelito mentions.
That model of Ibanez uses six individual bridge saddle pieces for a fairly broad range of intonation adjustment. Unless the strings are extremely old, or so brand new that they haven't stretched-into shape yet, you should be able to achieve your goal of Drop-A#.
Here's how you fix it: Experiment with turning the set screw fairly far left and/or fairly far right as you endeavor the intonation. Every now and then even the most experienced guitar player finds that he is turning the set screw the wrong way. Play around with it, starting with the middle strings (the D or the G). Make things go really sharp out of intonation, and then patiently go the other way, and bring things down to flat. And then so, right up to perfect.
It's only by pushing your left and right limits (sometimes) that you can find the proper middle.