There are various methods for changing strings, and there are a lot of strongly held but wrong opinions among guitarists as to which is the “best” way.  I have tried them all, and have found that the "Taylor method" is by far the easiest, fastest, cleanest, and most tuning-stable of them all.  LINK HERE for a tutorial from Taylor Guitars on how to correctly install strings.

I do have one small improvement to the Taylor Guitars method that is worth mentioning.  On the lowest/thickest two strings (6E, 5A) I cut the string 1 peg space past the string's peg like the video says to do.  On string 4 (D) I cut it 1.25 peg spaces past the string's peg.  On string 3 I do 1.5 peg spaces.  On string 2 I do 1.75 peg spaces.  On string 1 I do 2 peg spaces past the string's peg.  This gives all the strings' windings a similar distance between the hole and the bottom of the peg, and places the last wind below the hole.  This looks nice, and it also helps prevent string breakage, especially on strings 1E and 3G, by preventing string bending on the sharp edge of the hole while tuning.

Any time you change to a different brand or variety or gauge of strings, it is a good idea to check your neck relief (i.e. truss rod adjustment).  That is because different strings place different amounts of tension on the neck.  LINK HERE to see how to measure your neck relief and adjust your truss rod.

While you are changing strings is the ideal time to do some other simple maintenance on your guitar.  LINK HERE for details.