You don't have to spend that much, and if you are just starting out, it makes a lot of sense to go for cheap initially, especially if you will be using conventional steel tools. The reason is that in addition to leaning how to turn, you will also need to learn how to sharpen, and the process of learning how to sharpen tools can consume a lot of steel. So it makes sense to learn on cheap steel, and then after you have acquired some skills, upgrade if necessary.
Also, while it might appear to make sense to purchase a 'set' of tools, this can also be misleading because until you have learned something about turning, you won't know what kind of turning you enjoy most and therefore which tools you actually need.
My suggestion is to start out with an inexpensive 'house brand' tool - Benjamin's Best (the PSI house brand) is actually quite good and doesn't cost a fortune. The basic starting kit should be something like (and everyone has a different opinion here):
- a basic spindle gouge, either 3/8" or 1/2"
- a basic skew chisel (1/2" or 3/4")
- a basic diamond parting tool with a 3/16" - 1/4" width.
You should be able to get all three for around $100. And those three tools will enable you to try out a variety of different styles of turning (both spindle and faceplate).
Having said that, let me offer two seemingly contrary suggestions.
First, if you can find used tools, that's a good way to get started. I would expect that the most likely scenario for finding used tools is someone who is getting out of turning and is liquidating his/her shop. If you can associate with a local club, they may be able to point you to someone.
Second, a lot of people have gotten started using the basic tool set from Harbor Freight. These are not great tools, but they are good to learn on (learning both turning and sharpening), and many experienced turners still have and use that initial set. And often they say that having advanced to better tools, they still hold onto the Harbor Freight set for use in creating shop-made special purpose grinds.