Do I need both a lathe and a mill?
Yes. (And no.)
Either a lathe or a mill by themselves are limited tools. They are much more useful together than individually. It is true that some parts can be made exclusively on one machine entirely excluding the other, but it becomes very frustrating quickly to have to work with such a constraint.
That having been said, some lathes can be converted to mills with the addition of a milling column- so that the machine can be easily converted between 'lathe mode' and 'mill mode'. This gives the functionality of both machines at the expense of some convenience and some size capacity (working envelope). However, it can be a very cost effective way to purchase equipment to start machining.