Should the Z-Axis Home position be up or down? I have home-built CNC router.

depends, usually it is up to clear your work as you dont want it to move down into your work while referencing.
 Usually the highest off the work that your Z can move is machine zero and that means all z moves are in a negative direction with machine coordinates. Obviously when youhave work offsets you can have + and - z moves but in relation to machine coordinates they will all be negative moves.

Hood

Let's say that the home position on my CNC machine is 0,0,4 with the Z-Axis carriage moved up. When I run Cut2D program to create signs, etc, and load G-Code into Mach3 program, what should I do so that the CNC machine routs letters 0.25 inch deep on 0.75 inch piece of wood placed on the machine's base?

OK the normal way is home your machine, the zero position in machine coordinates for Z would be the highest point and the lowest would be Z-*(* being your amount of Z travel)
. When you want to machine something your code would have z0 as the top of the material and z-0.25 would be your depth of cut. To get that to work you would jog down to your material and zero the Z DRO, this makes a work offset of the surface of your material as zero and z-0.25 would be a cut of 1/4" into the material.
Hood