Hi,
I agree with others about the wisdom of buying second hand. Most lathes are available and a good choice would be a Boxford AUD or similar Myford. They usually arrive with a good amount of tooling which you will need. Remember that tooling for a machine can easily cost more than the original machine itself. A good second hand one will retain it's resale value and can often be sold at a small profit if yiou need to move on. Most of this smaller machinery has been used in schools where it will have had little wear, it mostly suffers from dings and paint wear where students have made minor mistakes.
The same applies to other machines such as millers where good second hand ones are available. For example, i managed to obtain a little used milling machine, the equivalent of the Warco WM18 (here) including a stand, Coolant pump and coolant, sets of end mills and slot drills, collet chuck and collets as well as a very nice small rotary table with a set of dividing plates for way less than the cost of a new machine without tooling.
It certainly pays to look around, but take a knowledgeable friend along as well. It may be worth visiting or joining a local ME club to get a bit of advice and experience – that can save a fortune.
Best regards
Terry