You have an inch plant, Tradescantia which is one of the easiest plants to grow.
Firstly your cuttings need to be in water, not just on the damp paper towel. I would cut those until you have the leaves with a 3 or 4 inch piece of stem, and put them in the glass half filled with water (change it every few days). In that time your cuttings will have rooted. At that point you can pot them up into a 3 or 4 inch pot - you don't want something too big at first, and use a houseplant compost which you can buy at the store or at a garden centre.
You can pinch out the growing tips as the plant grows, this will keep it bushy.
Good luck and enjoy your plant!
