Here's what I do. The caveat to this is just that, it's the way I do it. If any of you have input or suggestions or want to correct my procedure, please do so. We're all in this to learn and to enjoy this great sport.
After removing the spark plug and the cylinder head bolts, the cylinder should slide off easily. As the ring groove gets close the bottom edge of the cylinder wall, be careful and keep it straight. Wiggling it back and forth could scar the piston or cylinder wall, twisting it is ok if necessary. If it's an engine with quite a bit of run time, or has been sitting for awhile, reinsert the piston and use a good quantity of light machine oil to ease the process. I've got a can of Singer Sewing Machine oil that I use just for this purpose.
After you have it apart, take the ring out of the piston groove. Unless I want to keep the ring (I'm keeping this one just in case I need it in the future and because it's brand new), I don't care if it breaks, so I'm not particularly careful about taking it out. In this case, it was so much bigger than the piston diameter, it was easy to save.
Putting the new ring in is a matter of patience and paying attention. I've never broken one putting it in, but others I know have not been so lucky. The first thing I do is take my oil and fill the ring groove, making sure to get lots on the edge of the piston between the ring groove and the top. Then I pour oil all over the ring. You can clean up your mess later. I always check for the pin location and make sure I have ring right side up. You can tell by looking at the position of the pin and the relief points cut in the ring. The surest way to break a ring is to put it in upside down then have to take it off to turn it around. I start at the pin an insert one end into the groove. I apply a little pressure to the opposite end to force the ring open ever so slightly as I work it around. Keep the ring as parallel to the ring groove as you can while working it into the groove. When you get about half way around, it should just kinda snap into place. DONE, well not quite. You have to put engine back together.
Here's another opportunity to break the ring; inserting the piston back into the cylinder. Take your oil put a generous amount on the cylinder wall (all the way around), do the same for the piston. Squeezing the ring with 2 fingers, slide the piston in and the ring should follow. I never used anything harder than my fingernail to compress the ring and have always had good success with very little effort. The last thing is to use high temp thread locker on the cylinder bolts. Reinstall them like you would lug nuts on a car wheel. I usually use about 5 different steps in tightening the bolts. The first is barely seated, then go to the opposite bolt and again just barely touching the flange. Each time around the head is a little tighter until the last time when I really give it what for! A torque wrench is probably a good idea, but I don't have one of the appropriate size. And there you have it - my way of changing a piston ring.