The little nubbins are roots that are trying to grow. Cutting them off the potato separates them from their source of nutrition, so they will probably die. On the other hand, left attached to a potato (or a piece of potato, as @marinelife suggests) will be conducive to growth. The darn things grow in my compost all the time!
The tricky thing about potatoes is that the useful tubers form between the part you plant and the top of the dirt. So if you want to get very many, you have to plant them pretty low and keep heaping on the dirt right over the leaves. There are some pretty ingenious schemes for doing this, such as starting one in an old tire, and piling more tires on top (with more dirt inside) so at the end of the season, you have a tower of potatoes.
Also, a lot of the potatoes bought as food at the grocery store have been treated to prevent too much growth, so if you plant them, you may be disappointed in the results. If you seriously want to grow many potatoes (production amounts rather than “home science” curiosity), it’s best to buy “seed” potatoes at a farm or garden store. (Seed potatoes are small tubers that are probably already sending out some roots.) They’re usually inexpensive, and as someone who has tried both, I can say the results are better.