Yeah, pencil lead = graphite. Fan-fucking-tastic lubricant. Sorry if it's too "cheap" and practical for you.You wouldn't put pencil 'led' in your car's engine, would you? Pencil 'led' is not a 'lubricant'. Use nut slot lube, or even a very small drop of clean motor oil will work.
(You guys and your 'pencil lead'....)
A small drop of clean oil is mighty cheap too....although it is difficult to write with.
1. String your guitar properly. The best way is to under-wrap: put the string through the tuner hole, with enough slack to pul it 1/2" to an inch from the fretboard. Wrap the string 1/2 of the way around the post, the opposite way the tuner will wind. Pass that tag end under the main string, and pull it straight away from the peghead. Tune to pitch, you should end up with 1 and 1/2 to maybe three wraps if you did it right.
2.As already mentioned, stretch your strings. I hold the guitar in my lap, grab the string in my fist near the neck joint, hold the string in the nut with my other hand, and twist pretty hard, pulling the string straight away from both the head and bridge simultaneously. It'll take 4-6 stretches to get pretty close.
3. Always, always tune up, from flat to sharp. If you're sharp, tune down, and then back up. Other wise, the guitar will quickly go flat.
4. Learn to read your tuner. The initial attack is sharper than the sustained note. I tune to the sustained note, after about two seconds, the note should have settled down.
5. Lubricate and/or cut the nut properly. The stings should not "pink" during tuning or bending. If a little pencil lead or commercial lubricant doesn't fix it, the nut will need a taste of file.
6. Same as above, with the saddles.
7. Don't overlook intonation. If the intonation is hosed, you'll tune to open strings, find that chords are out of tune, retune to fix them, find you're out of tune in a different key, and then eventually do something silly like replacing the tuners.
8. It's never the tuners.