Ok I have an Epiphone Les Paul Custom Plus that has had a fairly bad tuning problem lately. The G-string keeps falling out of tune almost instantly after tuning it up. It's not even a small amount that it drops out, it's close to a quarter step.

Is this a string problem, a tuner problem, a bridge problem, or something else?
how old is it?
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its probably a mixture of a tuner problem and a nut problem.
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Bring it to the shop where you've bought it and ask them! If it's only a year old, it should still be under the .. garantee.. or whatever it's called in english! :P
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anyways, it could be a number of things. if you don't have experience in guitar repair and stuff you should just bring it into a shop.
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If it isn't a stretch problem, maybe the screw in your G string tuner is a little loose. Get a screwdriver and see if you can just tighten the screw on your g string tuner a little bit. Try that if your strings have been amply stretched.

I had the same exact problem and string. I just rubbed pencil lead in the nut and it has been fine ever since.
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99% of tuning problems come from the guitar simply not being re-strung correctly. Even the cheapest guitars in the world will hold tune most of the time providing that they are strung right.

Re-string the guitar, making sure to properly wear out the strings first. Then, do play them a bit. No strings, no matter how stretched before hand, will hold tune perfectly the instant you put them on the guitar. I find I need to play for about an hour solid before the strings - even ones that were stretched first anyway - truly hold tune.

If that fails, have your guitar properly set-up at a store. This should almost certainly solve and problems you have. I will honestly be amazed if any guitar over the £60/$100 price range still goes out of tune after a proper set-up and re-stringing.
If you haven't changed the strings in the past few months, do so.
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