Hi I've started many plants by cuttings. Ivy, fushia, begonias, impatients, geraniums, are all good candidates. Take about a 4" cutting from the parent plant, strip the bottom part of its leaves, leaving a few leaves at the top.
Where the leaves were attached at the stem, these are called the nodes, this is where the roots will form. Some use rooting hormones to stimulate roots production. I sometime use it and sometime don't. You will need a soilest mix with lot of vermiculite or coarse sand. Insert your cuttings pass the nodes, and water well, and make sure, you have holes in the bottom of your container for drainage. Your soil should be moist and not soggy wet. Depending on the plant you use, it could take 2 to 4 weeks for roots to form. keep soil moist at all time. You can cover the cutting with a plastic bag loosely, to keep in moisture, but it's not really necessary. hope this helps with your project.

Spider plants are easy- they grow baby plants at the end of runners.
Tradescantia is very forgiving. This and impatiens (Bizzie Lizzy) can be rooted in water so they can see progress.
Some of the cacti are very easy. Christmas cacti can be propgated by breaking off a couple of segments and planting those. Hummock forming cacti like Rebutia are also very easy. Wear gloves, break or cut the babies off the parent, allow to dry for a couple of days before planting in sandy soil.
Or what about ornamental ivies? They will root in water or soil very quickly.

You could also try a pothos. im always making cutting of mine and rooting them in water. They seem to root fairly quickly and i have never had one die yet.

coleus impatiens or begonia [not the bulb type] .... which will be frosted down in a few weeks outdoors .. can be rooted in water ... or in a small indoor greenhouse [take home food thingee] ... or a baggie over a pot ...

use any of the above names with 'propagation' and enter it into google ... for millions of 'how to's'

ken

Another thing to try is succulents.
Many of them will grow a new plant from one leaf.
Just pull off a plump healthy leaf, and lay it on some sandy compost or even just on a dry plate. Very soon it will start to grow roots, and then a new little plant which can be potted up. Eventually the old leaf will dry up and die.
eg Echeveria