That's what I use too; just an old cotton or linen rag. Even a paper towel will work, but a nice soft cotton rag works best. You want cotton because plastic fabrics like acrylic or polyester will just melt on the hot pipe and make a mess.

And, for removing excess solder from a socket, I just take a piece of copper pipe, sand and flux the end of it, heat the socket and insert the pipe end. The molten solder inside the socket bonds to the pipe end so that when I pull the pipe end out of the socket again, there's just a nice tinning of the socket with no excess solder in it. (Then I just flux that tinned socket, sand and flux the new pipe end, insert the two together and solder normally.)

And, ALWAYS ALWAYS ALWAYS when soldering, keep a spray bottle full of water handy so that you can put out a fire in a gawd awful hurry if you need to.