You are asking great questions.
The path from an idea to a finished project has multiple steps. First you have to draw the part in some kind of Computer Aided Drafting (CAD) program, then you run it through a Computer Aided Manufacturing (CAM) program, then apply that output (G-code) to a Computer Numeric Control (CNC) program that actually runs the machine.
I think ArtCam will handle both the CAD and CAM functions. ArtCam outputs G-code that a CNC program can understand. But you still need the CNC program. Mach3 is one option for a CNC program.
First you need to define what ''high precision'' means to you and decide what is an acceptable level of error. No machine is perfect, and the cost of the machine goes up as function of roughly 1 / the square of the precision. 0.01'' accuracy is relatively inexpensive and in the range of most hobby class machines, many are a bit better. Consistent 0.001'' accuracy is going to be expensive and is in the range of heavy industrial class machines. Now it's a matter of balancing your budget against the precision that you want. I usually describe it like this: Accuracy, Speed, Cheap; choose any two.
Now you have to decide what drive system you want on the machine, Open loop stepper (least accurate, least expensive), closed loop stepper (better, slightly higher cost), full on AC servo (most accurate, most expensive). Ball screw drive is generally more accurate than rack & pinion or timing belt drive.
There are two main type of machines, moving gantry and fixed gantry. In the case of the fixed gantry machine, the table moves in at least one axis rather than the gantry. These are generally more rigid machines and thus more accurate, but require larger axis motors and are more expensive. They also take up more floor space, an 8 foot machine would be about 14 feet long.
The higher end machines normally come with their own CNC software. Many hobby class machines do not come with any CNC software, some are supplied with a trial version of Mach3, no computer, but are ready to connect your computer. Mach3 has a lot of online support through user groups and forums like CNC Zone.
I hope this helps get you started.