I use an oilstone to tune up my tools between turning sessions. Periodically, I need to take more drastic measures, and for that I use a sanding disk.
My lathe came with a faceplate. I glued a piece of MDF to a piece of plywood, mounted it on the faceplate, and turned the whole thing down to a circle slightly larger than the faceplate. Then I glued a piece of sandpaper (emery cloth would be better) on the MDF.
To sharpen, I just mount this on the lathe, run it at the lowest speed, and sharpen my tools. I usually hold the tools against the toolrest with blade at a downward angle so that the bevel on the tool rides against the face of the sanding disk.
I got a starter kit with my lathe that included three tools (gouge, skew and parting tool) that are made of tool steel. These need to be sharpened very frequently. I've since bought some HSS tools that don't seem to get dull nearly as fast. But the cheap tool steel tools work just fine if they are sharp, are much less expensive, and have helped me learn how to sharpen tools.