While a mill may be very handy; in stuff like cars, motorcycles etc, a lathe has more direct application to things shafts and bushings. All precision machines trace their roots to the screw cutting lathe. While some CNC machining centers can do things like thread milling and O.D./I.D. boring a manual mill is less adaptable to turning work.
A mill can be one of several types of machines; vertical, horizontal or combination. The mill most people think of when you mill in garage settings is a Bridgeport. These are very handy machines but have several limitations. In any machine tool weight and mass are attributes that directly correspond to rigidity and performance. A mill like a Bridgeport J-head is a light milling machine, they work well with small cutters, light feeds and moderate cuts. The bigger machines of the same design are 1 1/2-3 times the mass of a Bridgeport.
A lathe is like a mill with small hobby or sizes up to battleship grade machines. A small lathe like a South Bend or Logan will do light work, a bigger heavier lathe will do much of what the little one does and have more power and mass to make heavier cuts on tougher materials.
Purchase prices are all over the place and space is a major consideration when thinking about machine tools. Space is perhaps the biggest issue for many because machinery can take up a lot of real estate. How much space do you have for machinery? This should be where you work backwards from. Then what do you envision doing? Then narrow down what kinds of machines you need and select a few makes and models to research. When buying any machine tooling is a major issue. New machines or used machines with no tooling end up costing far more than the sale price and you could easily spend the price of the machine on tooling.
There are no quick easy answers to should I buy a lathe or mill? Define what you want to do, decide if it is practical with your skill level, space and budget. If you have few skills take some classes at a Vo-tec or Community College before you make any choices to get an idea of this is for you or not. It takes dedication to learn the skills and become proficient let alone be safe. Don't waste you time and money on something that you are not at least safe to operate.
Steve