I am fortunate in that I own my own trucks, trailers, and heavy equipment for load handling, so handling something like this is not an issue for me.
If I were in your shoes, I would arrange for the lathe to be shipped to your local freight terminal and pick it up there. Inspect the crate closely for damage before signing the acceptance form, and note any damage observed on the crate on the forms. I would use a truck and a small trailer (the smaller the better so that you can have positive tongue weight yet still fit the lathe under an engine hoist).
Most trailers will handle a 1,500 lb load, so your issue will be in manipulating the load once you have it home. As others have mentioned the lathe is top heavy; when I brought my recent 12 x 36" lathe home I used a pair of 3" nylon web straps to pick it up by the bed (picked up by hooking the straps to some skid steer forks). Handling the lathe suspended from the straps is safer than trying to pick it up from the bottom (won't tip over on the straps).
An engine hoist is an option (and you can rent them), but they can be a bit "tippy" too and the small wheels don't roll well across a rough surface. It's better to back the trailer with the lathe into your shop, open up the carton, and use the engine hoist to lift it straight up. Then pull the trailer out and lower the lathe until it is almost touching the floor before you attempt to roll it around on the hoist.
Chssesking's post with pix shows an excellent way for an individual to handle this.