I'll try with bullet points! lets go!
fist cut your old cable, get some nice dikes or something to just snip but take note of how it runs through the bike, how its routed. you're ganna wanna follow the same path.
remove the housing off both ends, toss in garbage
lets move to the front of the bike,
on your brake lever, there is a little brake nipple, this guy!
. solid part on top. split on the bottom of the lever, and the cable runs through the slit
what you wanna do here is push the cable INTO the lever more, and the cable should push back into the lever through the nipple. this will allow you to remove the nipple by pulling it out through the top sometimes a flat head pushing on the bottom helps a lot.
now that the nipple is free, don't loose it, throw it in your pocket.
take out any threaded adjuster from the handle, your new cable (as long as its a real moped cable and not a bike cable) will have a new adjuster on it for both front and rear.
the handle bar end of your new cable will be kinda bulbous and the end for the wheel is more of a sideways cylinder shape. so, now grab the nipple from your pocket ( haha) and make sure you have the right cable end. the rounder bulbs one. pull on it to get some slack, and pull the lever, you can stick that bulbous part into that handle, then put the nipple back and whammy it should lock the cable in! now put the adjuster in the adjuster hole ( the new one may not be threaded but should fit in)
now, make sure BOTH adjusters are set allllll the way in so they can extend the max amount. and give you the most adjustment once its on.
route it to the back wheel!
so now to remove the back portion its pretty straight forward the rear adjuster looks like a long hollow bolt with two nuts, remove the jam nuts and thread the adjuster out. cable should come off easy.
new rear adjuster should looks the same as the old one, (a hollow bolt with two nuts) put it back in!
that sideways cylinder bit goes into the slot on the lever thats on the brake plate, remember you want that hollow bolt alllllll that way into its threaded holder.
now start to tighten that hollow bolt adjuster, TECHNICALLY you're backing it out of its hole, but you'll see its also pulling the brake lever slightly with every turn,every every couple turns of the adjuster, try and move your rear wheel. are the brakes dragging? if not keep going.
tighten until they drag then back off just a little, now, you're probably going to back that hollow bolt out most of the way and its still not dragging, that's okay, when you get the hallow bolt out 3/4 of the way, move to the handle bar adjuster and start doing that one a little maybe half way. keep turning your rear wheel and checking it.
when you get it good and tight, use the jam nuts on the adjusters to lock them in place. your wheel should not drag when spun but be close to it. to jam the rear, tighten both bolts against where the hollow bolt threads in, one at a time, for the front one at the lever, jam the nut against the spinning cylinder part that you've been twisting to adjust it out.
hope this helps. just look at it a bunch and take not of how stuff works as you're taking it apart it helps with putting it back together again. (edited)