I started turning just about 4 years ago and jumped in with a starter set of tools and upgraded since then. This was about 7 lathe tools in a starter set and I had to also buy something to sharpen them with.
I have a new recommendation that has come to market in just the past couple of years that I think will encourage new folks without getting into tool overload.
Here is my is the Tools you need 101:
2) A parting tool. You need one, preferably narrow kerf, end of story full stop.
3) A multi bit tool interchangeable cutting head on a tool. Circle, Square, Diamond point.
4) Diamond sharpening plate.
A few reliable brands: Rockler with their house brand, Easy woodworking tools, and Robert Sorby (others?).
Each tool has a set of Pro/Cons. I think you can get down two 2 tools if you go with the Robert Sorby solution. If you choose the other you can do it with 4 tools. This is enough to pretty much do 90% of woodturning; hollow forms being the need for more specialty things, and threaded boxes.
HSS (High speed steel) and Carbide do not have the same grain structure they behave somewhat differently and there are pro/con to using one over the other. Options for both on one tool are fantastic.
The big upside is you don't need to buy a sharpening system from the get-go (grinder, or belt) but rather just a diamond lap plate is sufficient. You remove the cutting head from the tool and just lap the cutter head on the diamond plate using water for lubrication.
After you start you can always add more tools ;)
PS. If you get into turning buy a 4 Jaw Chuck (number 5 on my list).