First, putting the rings on the piston can be tricky, since the compression rings are cast iron and if you just pull them apart to expand them, they may break. There is a tool called a ring expander to eliminate this risk, but while it is probably essential for deep, high-tension rings like those on many diesels, it isn't really needed for a lawnmower engine. Just remember to pull them gently wider at the gap, not halfway across the ring, so the bending deflection will be somewhat distributed around the ring's circumference. Expand them barely enough to get them over the piston, and treat the cast iron as glass: it isn't much more flexible than that.
Second, having put the rings on the piston, you have to compress the rings to get them into the bore. With anything larger than a lawnmower engine this calls for a ring compressor, but for the low-tension rings lawnmowers use, some people just compress the rings one at a time with their fingers. Most of us regard that as hard work, and prefer to use an ordinary ring compressor you can get from a speed-shop. Remember though, if you are working on an engine that has big end bearings that are not easily dismantled (that is, a one-piece connecting rod) you have to put the piston in from the bottom of the bore. That means the ring compressor ends up looped around the connecting rod after the piston is in the bore. That is OK if the ring compressor can be opened up to make a gap in its circumference, but most of them can't. For one-piece-rod engines, you usually use a piece of sheet aluminium with a twitch of tie wire around its outside, as a ring compressor. When the piston is in the bore successfully, you cut the tie wire and open up a gap by partially unwrapping the sheet of aluminium from its circular shape.
