To take off the fine scratches and burrs left by coarser stones, and to polish the surface, you can use stones starting at around 2000 grit. There is theoretically no upper limit, but stones above about 10000 grit achieve practically no measurable improvement in the edge. It is also interesting to note that above 8000 grit, there is no Japanese measurement standard. For stones labelled as having a finer grit, you simply have to take the manufacturer's word for it.
Lubricate the stone. Some stones specifically use oil or water, and if that's the case, ensure you're using the recommended lubricant. Most importantly, whichever lubricant you choose, do not change it after the first use. When using oils, only use those approved for sharpening stones. Food oils such as vegetable and olive oil should never be applied! Some options like diamond stones, and others, don't need any lubricant at all, so be sure to check the stone's instructions.
A single stone of 120 grit and a combination stone of 1000 and 3000 grits come along with a stone holder all for a price of less than many other individual stones. The stones are 6 7/8" long and 2 1/8" wide. A flattening stone of some kind would be needed, but with economical options available in those, the overall price of this kit would still be low. This entry level set is a good budget minded option.
Begin with your lower-grit stone. Place the heel of your knife on the far edge of the stone, holding the blade gently but firmly with both hands at a 15- to 20-degree angle. Using even pressure, slowly drag the knife over the stone toward you down the length of the stone while simultaneously moving the knife such that the contact point moves toward the tip of the blade.
Steeling helps maintain sharpness. This process realigns the edge, correcting for dulling causes such as a rolled edge. A sharpening steel is a type of hardened cylindrical rod used similarly to honing stones. For example, a butcher steel is a round file with the teeth running the long way, while a packer steel (used in the meat packer's industry) is a smooth, polished steel rod designed for straightening the turned edge of a knife, and is also useful for burnishing a newly finished edge. Because steels have a small diameter they exert high local pressure, and therefore affect the knife metal when used with very little force. They are intended for mild steel knives that are steeled several times a day, but are not well suited for today's tougher and harder blade steels. Diamond steels are now available that have an industrial diamond coating and can remove blade metal as well as straighten, therefore used correctly they can re-profile a knife instead of just honing.