I treat restringing my guitar as a bit of a meditative event. It is important to get things right and not rush too much otherwise you will pay the price later.
I generally remove all the strings at once, give the fretboard a clean, then put all the strings one and leave them quite slack until all 6 strings are on with just a little bit of tension on each. Enough tension to hold them in the saddle and nut slot and keep the windings tight on the post.
Then I tune them up fairly quickly, and I usually tune them up to a semitone higher than normal (e.g. E to F etc.). Once done, I play a few riffs making sure I play on all 6 strings and do a couple of whole tone bends on each string. Also play a few bar chords up and down the neck using a heavy pick action.
One thing is to ensure that the strings go around the tuning post at least TWICE - sometimes more. Also, important to make sure that the windings are neat and do not overlap at all.
Oh, and I usually also run a very sharp 2B pencil in the nut slots before I put the new strings on. This helps the strings to glide through the slot easily. Careful though, if the graphite smudges it can look messy.
After the first play through at higher tuning tension, I will grab each string at around the 12th fret and lift them off the fretboard a couple of centimetres. Give them a tug basically to ensure any slippage will be taken up. Then I play a bit more with a few more bends and chords for about a minute.
Then I detune it back to normal tuning. It is important to detune BELOW the right string note and then tune back UP to the correct note.
After all 6 strings are done, I play for another minute or so, and I generally find that in 90% of the time, the guitar pretty much stays in tune after all that without too many major adjustments needed.
Hope this helps.
JBLP Gold Top #129 - redubbed "#1 in Oz"