If this seems like an odd combination for a subject name, then you might be at the right place. That's because, unless you use a vegetable peeler (which I do not), you need a knife for both peeling and cutting up potatoes. I like a paring knife for peeling and a butcher knife for cutting up. Having the right size knife, heft if you will, is almost as important as having a sharp knife. You can get by with a dull knife but it makes a job out of something that is usually quick and easy. And having a sharp knife is easy! You can sharpen one with the rough bottom of most cups/mugs if you don't have a sharpening stick. Hold the knife at an angle and place the blade near the handle against whatever you are using for a sharpening stone. I'm sure there is a preferred angle but any below say, a 45 degree will work fine. More important than the degree at which you hold the knife is that you get pretty close to the same angle every time. Otherwise you will tend to roll the edge. Pull the blade against the stone down the entire length of the blade. Repeat at least two more times. Turn the blade so you can do the other side and repeat the process. You will be surprised how much this will help even a dull knife and it will keep sharp knives sharp. A couple of caveats - if the knife is very dull, repeat the process a few times. The second caveat - this is for inexpensive knives. If you have a nice knife or set, then it's probably a good idea to have it professionally sharpened.
One more thing about knives - always run water over the knife after you peel or chop anything! Some things may not be harmful but lots are. This is especially true of lemons and tomatoes. They will cause the knife to rust and it won't take long for this to happen. In fact anything in your kitchen that is metal does not need to set for any length of time after being exposed to fresh fruits and vegetables. This includes grinders and such that have a chrome finish.
Now about those potatoes...
I cut them into slices about 1/8 to 1/4 inch thick. They will cook very fast this way and you won't have hard spots as you sometimes can if you cut them into irregularly shaped pieces. It is not absolutely necessary to slice a side off the potato first so it sets solidly on the cutting board but it's a good idea. I think quite a few people have cut themselves using a knife. I never have. Just remember you have a potentially lethal weapon in your hand so not a good idea to play fast and loose with it.
Put the potatoes in a pan and cover with cold water. Add a pat of butter or your favorite grease or oil. The reason for this is not to add to the taste so much as to prevent the potatoes from sticking and also this will help about the pot boiling over unexpectedly. Put a lid on the pot and turn the heat to about medium low. Yes you can cook them without a lid. It will take longer and use at least twice as much energy but some people do it that way. You only want to leave the heat on medium low until the water in the pot gets hot. At that point turn the heat to very low or simmer. The heat should be low enough so the pot will not boil over even if you leave it unattended, which I do fairly often. At some point after the potatoes get hot you should add an equal amount of salt and sugar. I don't try to actually season to taste at this point - just begin the process. It will not take long, around 30 minutes, for the potatoes to cook. The potatoes are done if you peck one of the slices with a spoon and it breaks apart. Take the pot off the stove and drain the liquid. Paula Deen says she sets the potatoes back on the stove and lets the excess liquid cook off. I think this is a good idea but it is very easy to burn the potatoes and easy, at least for me, to forget to turn the stove eye off.
You can add several different things and combination of things to the potatoes before mashing them. I like whole milk (actually that's the only kind I ever buy) or buttermilk. I'm sure half and half or cream would taste good but is probably unnecessarily fattening. You can add a mixture of milk and soy milk! I would be careful about the soy until I experimented to see how much you can use without adversely affecting the taste. You will need to add one or the other - mayo or sour cream, whichever you prefer or have on hand. If you don't mind washing a potato masher use that now or if you want the potatoes whipped, dust off the electric mixer. I like the potatoes just a tiny bit chunky so I use a table spoon to mash with. Besides I don't really like washing a potato masher.
Add salt to taste. That's it.