i've never used a cable puller for this. only brake cables.
To replace a derailer cable, you want to start out with both the derailer and the shifter in whichever position has the cable slack. This will usually be the position corresponding to the smallest chainring/rear sprocket.
If you disconnect the cable (or if it broke) and pedal the bike a couple of times the derailer will automatically go to the correct position.
For the shifter, you may need to pull on the end of the cable while operating the shifter to get it to shift to the loosest position.
Once the derailer is disconnected from the derailer's anchor bolt, pull the housing (if any) away from the shifter, and then push the expoed inner cable into the shifter. The moulded end of the inner cable should then pop out of the shifter. If it's a simple lever shifter, such as a down-tube or bar-end shifter, the cable end is readily visible.
If it's a more complicated ratcheting type shifter, there will usually be an access hole where the cable end can pop out...but his access hole may only line up correctly if you remembered to shift to the loosest gear position first.
Twist-grip shifters such as Sram GripShift are typically the most difficult for cable changing, and these commonly require disassembly. Someday I'll try to find the time to explain this process.
Generally the derailer limit stops should not require any adjustment when you replace a cable, but you will need to adjust the indexing.
Before connecting the cable, screw the adjusting barrel all the way in, then back it out maybe a turn or a turn and a half. This will give you the opportunity to loosen the cable a little or to tighten it a little or a lot.
Front Cable Replacement
Thread the cable through the shifter and any housing. Check the condition of the housing and particularly the open ends. Replace or trim if needed. See also my Cables Article. Pull on the cable and operate the shifter until you have the slackest position. The chain will normally be on the smallest chainring.
Run the cable under the anchor bolt hardware and secure it. If it's an indexing system, particularly a triple chainring, you will need to fine tune the indexing to get it to shift well and run smoothly on the middle chainring.
Rear Cable Replacement
Thread the cable through the shifter and any housing. Check the condition of the housing and particularly the open ends. Replace or trim if needed. Replace or trim if needed. See also my Cables Article. Pull on the cable and operate the shifter until you have the slackest position. The chain will normally be on the smallest rear sprocket.
Use a pair of pliers to pull the cable really tight and secure the anchor bolt. Use the adjusting barrel to correct the indexing adjustment.
(This is how I do it. Only works if you have the bike in a work stand of some sort, unless you have three hands.)
Hand pedal forward with your right hand while manually pushing the rear derailer inward until the chain engages the 3rd smallest sprocket. Stop pedaling, then let go of the derailer. The derailer spring will try to move the derailer outward toward the smallest sprocket, but the stopped chain won't let it go all the way.
Pull fairly firmly on the end of the cable to take up the slack and secure it with the anchor bolt. Then you can pedal and check the indexing adjustment. I find that this usually gets me quite close, with only a minimal amount of fine tuning needed to the indexing adjustment.
Cable Attachment (rear)
One area that commonly causes problems is the attachment of the inner cable end to the derailer's anchor bolt. If this comes in at the wrong angle, it can change the geometry of the parallelogram and make it impossible to get the derailer to index properly across its range. There is usually a groove in the derailer body or a washer with a bent corner to determine that the cable is running correctly.