I start with the 6th string and change them 1 by 1, tuning and stretching the new string before I cut the next one out.

kind of depends on what kind of rig you have...if it is a string throught body type it will be strung differently than if you have a kahler or floyd rose tremolo set up. regardless of the kind of guitar you have, you should remove all the strings and clean the guitar. Much oil and skin and gunk accumulates on the neck and such of the guitar from playing it. This gunk can affect the intonation and sound of your guitar. So clean it up really good after you remove the strings. If it's a standard tremelo, string through the body type you have to remove a panel on the back of the guitar and run the string through the body out the front and then attach them to the machine heads at the head of the guitar and tighten them up. There's many theories on which strings to string up first...some say it doesn't matter, some say you sould put the low E on first and then each higher, others say to put the highest first, others say put the highest then lowest and alternate between high and low till you put them all on..I go with the last option since it seems to keep the tension better on the neck and strings seem to slip less. After you have put on each string, and tightened it (remember do not tighten each string to key they will lose tuning anyway as you add other strings) just get the string on and wind it a bit. Don't try to wrap the whole string onto the machine head/tuning pin, too much string is just more string to slip, you should have a good 4 inches at least of extra string that you will clip at the end. If you have a kahler or floyd rose tremelo you will not run the strings through the body. You will have to get some wire cutters and clip the little "O" ring off the bottom of the new strings. Then on the back of the tremelo you will see 6 little hex screws. Loosen these (only after you have turned the tuning keys and loosened the strings till they are flapping) and you will see the where the strings are coming from there is a little metal square that is pinching the string. The string will come out and you can remove it. Then take your new string with the cut off "O" ring and place the end where the "O" ring used to be into that hole. Then turn the hex screw again until it is tightly pinching the string and it will not slip when you string the guitar through the tuning pin and tune it up. Use whatever order you want to start attaching the strings and tightening them up/tuning them and when they are all attached you can start going back and do a final tighten and tune the guitar. I use and prefer kahler and floyr rose tremelos because they can be tuned better and have a fine tune knob on the tremelo as opposed to a regular tremelo which you can only tune by the tuning keys..and defeats the purpose of having a tremelo locking nut. Also with kahler and floy rose should you break a string often times since you have already cut the "O" ring off, you can just detune the string to get some slack and then reclamp it back into the tremelo and there you go, no need to replace the string.