every time I replace guitar strings, the strings will always relaxing for a few hours to become stable.
how do I speed up this process?
When removing the old strings DON'T take them all of at once! :)
Remove the old string, then replace it with the corresponding new string.
If you remove all of the old strings at the same time, then proceed to string the guitar with the new strings you will be greatly decreasing, and then increasing, the tension on the neck.
This is not good for the neck, and could lead to tuning stability issues concerning the new strings you installed.
I usually remove/replace the strings ONE AT A TIME in this order and find it has greatly cut down on my tuning issues.
A G e B D E
Down then back up again. Remember to *stretch each string and tune to pitch about 3 or 4 times before moving on to the next string. Then when the guitar has all the new strings on it, go back and tune the entire guitar from low E to high e.
*Stretch a string by gently pushing it towards the floor then pulling it towards the ceiling a few times with the guitar in your lap. Then hold down the string at the 3rd fret and pull that string out, away from the body a few times; rinse and repeat at at frets 5, 7, 9, 11, 13, 15, 17, 19, 21. Careful not to pull or push to hard or you WILL break your new string! :(
Stringing a guitar takes some time. Maybe 20-25 minutes including setup and cleanup time.
Just don't rush through it and you'll be fine.
Hope this helps to solve your tuning issues!
On guitars with Floyd Rose trems or similar, I just tune them about right, give them a good stretch by pulling back on the bar a few times and re-tune them.
That gets them pretty close so I just lock the nut at that point and any fine tuning can be done at gig time using the fine tuners at the bridge.
I used to stretch them in much the same manner as Alistair mentions, but I don't bother any more; I just tune 'em to pitch and deal with minor re-tuning. I find that steel strings rapidly settle in.
Also, you don't mention, but nylon strings take MUCH longer to settle....
When I replace my strings, I follow the same process.
I take the old strings off, and then place the new one on. After this, I tune it to roughly the right pitch, and then place my thumb on the top of string and push down, while underneath my two fingers pull it up against the pressure of my thumb, stretching the length of the string a fair bit. This trick helps stretch the string ready for tuning, so that when it is tuned it settles quickly.
After that, I leave it, and repeat exactly the same process for each new string. At the end, I tune each string on the guitar up to pitch. The strings stay in tune very well after that, perhaps requiring a very small amount of fine-tuning after about 5 minutes, but they 'settle' very quickly.
Hope this helps!