There are a wide variety of woodturning tools used to shape wood and many styles of faceplates and chucks and all of their accessories used to attach the project to your lathe. So start small and cheap until you know where your interest lies. Woodturning usually starts out as a hobby; as ability and desire develops, it will sometimes move into some form of business.
The general tool groups of cutting tools for the wood lathe are gouges, skew chisels, scrapers, and parting tools. Each of these groups of tools is divided into various subcategories as in gouges; There are detail gouges, spindle gouges and bowl gouges and these are further divided with different diameters and thicknesses of the individual tool. As you can see, it is very easy to acquire many different tools.
Tools must not only be kept sharp, but must also be ground to angles that fit the design of the tool. Most shops will have a variety of files, sharpeners, whetstones, low-speed beams, high-speed grinders, and abrasive wheels that work in water. Grinding wheels are made of aluminum oxide, silicon carbide, ceramic, diamond, and cubic boron nitride. The point of all of this is to show the wide variety of options and go slow when you start; work on these tools slowly. The best way is to make friends with an experienced turner.
Most wood lathe tool grinders use multiple jigs to grind compound angles, bowl gouging can be very challenging without a jig to control angles. As an example; the simple scraper should be sharpened at an angle of about 30 degrees; if the tool is sharpened at a greater angle, it will seize and will gouge your work. Many wood turners will use tools straight from the grinder because the edge of the wire on a sharp tool will leave a very fine cut in the wood.
All lathes, tools and grinders kill the quality of your project in simple pieces of sandpaper. There are a variety of sandpapers made from different materials and thicknesses. You can sand by hand, or use power sanders or pneumatic sanders. Be sure to consider dust collection, sanding will fill a shop with dust very quickly.
Woodturning is a fascinating art and I am not writing this article to discourage anyone, quite the contrary. I like to see other people go into woodturning; I just hope your ride is faster and smoother. There are several woodturning projects with pictures showing how to do them and there are also some technical tips, all on my blog.