If you’ve played electric guitar for very long, you’ve run into this problem.  Tune up, using a chromatic tuner and harmonic tuning by touching each string lightly on the 12th fret while plucking.  Everything is spot-on.  Play a G chord.  Beautiful.  Then, play an E chord.  Suddenly the G (third) string sounds out of tune — sharp.  Play another G chord.  Beautiful.  E chord.  Out of tune.  Argh!

This problem mainly occurs with electric guitars. There are a few things which combine to exacerbate the issue, and a few things you can do to mitigate it.

On acoustic guitars, the G string is wound. On electric guitars, it is not. This is one contributing cause. But since you can’t fix that unless you switch to playing acoustic strings on your electric guitar (not recommended), the solutions below are the main course of action.

1. First of all, when tuning any guitar or other stringed instrument, always tune UP to the correct pitch, never down.  If a string is too sharp, first tune it down until it is a bit flat, then tune it up until it is right on pitch.  This ensures that any slack is taken out of the tuning machine, so that when you bend the string, it won’t go flat.  Always finish tuning a string by tuning up to the correct pitch.
2. Most chromatic tuners tend to be a bit unstable and/or inaccurate when tuning with open strings, especially if your guitar has a really nice sound with lots of harmonics.  If you can’t beat ’em, join ’em.  As you’re plucking each string, touch it lightly over the 12th fret to produce a pitch which is an octave higher than the string’s open pitch.  Don’t fret the string — just touch it above the 12th fret while plucking.  A soon as you’ve plucked the string, remove your finger.  If you do it correctly, it will produce a sound like a bell, one octave higher than its normal pitch.  The harmonics from this will also be an octave higher, out of the range of the tuner’s pitch measurement capabilities, thereby increasing the stability and accuracy of the tuner’s pitch measurement.  This works with all guitars and most stringed instruments.
3. On some electric guitars, especially the less-expensive ones, the action tends to be too high at the nut. This can usually be fixed by either lowering the nut, or making the nut slots slightly deeper. Not so deep that it causes buzzing on the first several frets though.
4. You may be experiencing friction at various points along the span of the strings, which causes string binding and results in tuning instability.  For Stratocasters, I recommend having the Fender LSR Roller Nut installed, switching to the Fender staggered locking tuners (which eliminates the need for a string tree, and makes string changes a snap), and removing the string tree. This greatly improves tuning stability. For Standard Strats, you can also replace the stamped-metal saddles with third-party chrome saddles which are like the ones which come standard on American Deluxe Strats, which will greatly improve your tone and increase sustain. Search the web for “BP-2333-010”.  Changing the saddles requires a complete string height and intonation setup.  Also, increase the tension on your tremolo springs until your tremolo rests firmly (not too firmly) against the guitar body when the guitar is in tune, and doesn’t come off of the body with string bends. You may have to add a spring or two to make this happen. With just these inexpensive mods, you have the effective equivalent of a Floyd Rose tremolo system, without the pain of a locking nut and corresponding dual-tuning system. I much prefer this setup to a Floyd Rose system.
5. The frets are too tall, especially the first and second frets. A decent, low-action fret job can fix this. Have a luthier reduce the height of the first few frets, and adjust the remaining frets to avoid buzz.
6. Using too thin of a string gauge. For electric guitars, you should use at least 10s, for several reasons. The thinner the strings, the sharper the G string will go when you fret it on the first, second, or third fret. Also, thicker strings will give you a richer tone and longer sustain.
7. Tuning the G string to standard tuning. You should always tune your G string a few cents flat. Not so flat that it sounds flat when playing an open G chord. Test your tuning by playing an open G chord, followed by an E chord. Alternate between the two, and adjust the tuning of your G string so that a good intonation balance is achieved between both chords.  You can also play the G string open, measure where it is on the tuner, then play it fretted on the first fret and measure again, then split the difference so that it is flat by the same amount when unfretted as it is sharp when fretted on the first fret.  Once you find a happy medium, put a mark on your chromatic tuner’s scale (it will be just a tad to the left of center) where the G string registers when played open (unfretted), and always tune your G string to that mark instead of the center mark, while tuning all other strings to the center mark.  Then, say goodbye to that misbehaving G string!