The two
"two black plastic rings" are the piston rings. These rings have a special way of going on. There are a alignment pin that the ends of the rings align up with.
You being totally inexperience of installing a piston here is the best way for you, in fact the best way I've found for these engines, as it helps in not breaking rings for a noob.
1) Remove the cylinder (which you have done).
2) Remove the piston (remove "C" clip retainer from ONE side)
3) Slide out piston wrist pin (only enough to remove piston from piston rod)
4) Remove piston from engine. (note as to where the front is, mark if you need to. There should be a arrow/mark on top of the piston. This will be aligned with the exhaust port)
5) Place cylinder on work area, upside down (piston skirts upward)
6) Place top ring as needed (aligned with piston alignment pin).
7) Set piston into the cylinder, the ring will contact the cylinder base, the piston base is beveled out.
8) With your fingers squeeze the ring into the piston. The cut away in the skirt helps accomplish this.
9) Tap on bottom of piston (do not force) and slide the first ring into the cylinder.
10) Repeat steps 8 and 9 for the bottom ring.
11) Cylinder and piston can now be installed over the studs.
12) Align piston on piston rod.
14) Insert the wrist pin.
15) Install "C" spring.
16) Tap cylinder in place.
Before I install the cylinder I use a gasket sealer on each side of the gasket. I like...

There is a dabber attached to the cap...I cut it off and use a small paint brush as the applicator. This bakes for a neater/cleaner application, as any excess will not be the easiest to remove.