You have been given some good advice by others. I am throwing in a pennies
worth of information.

I bought the Harbor Freight (HF) lathe #34706, it retails for $279 and goes
on sale periodically for about 200 dollars. I bought a 8 piece gouge set
from HF, I believe I paid $42 and it now sells for approx. $36.

The lathe will serve me well for some time. Time, interest and my skill
growth will tell if I need to Invest in a second and larger lathe.

I bought some bowl gouges, a chuck, roughing gouge and skew chiesel from
Wood Craft.
I have done some spindle turning and have done 4 bowls (about 10 inch
diameter). You will not need these items in the beginning if you are going
to turn bats, that is you can get by with the HF tools. I was fortunate to
buy the Wood Craft items on sale.

If I were more interested in turning spindles the HF Gouges would work fine
as a starter set. This will give you a chance to practice sharpening on a
cheaper gouge, at least for me it is less intimidating.