If you are just doing spindle work, the minimum would be a spindle gouge (1/2" or 3/8"), a spindle roughing gouge (3/4" or 1") and a parting tool (1/4").
If you want to expand on that, you could add a 1" skew chisel but it isn't essential to start. You could also add either a square scraper (3/4" or 1") or a half round scraper (3/4" or 1/2") but neither are essential.
If you expand your repertoire into bowl turning then add a bowl gouge (3/8" or 1/2") but you would need a chuck to turn bowls.
You can buy a set or you can just buy individual tools. There are always plenty of second hand tools if you look at the usual places online. Good quality second hand tools would be better than cheap, low quality new tools. Stick to the well known Sheffield brands and you can't go far wrong.