Why do I have 110 volts from the neutral to either line on a 220 plug, but when I test across both lines where I should have 220, I have nothing at all?

Probably because our feeding from the same phase. (120-120) =0. Your meter does the subtraction.

Vmcos(theta)-Vm)cos(theta) =0

vs.

Vmcos(Theta)-Vm(cos(theta+180)) = 2Vm

Please describe the breaker you are using to get your 240 VAC.

Normally you should find a tandem( 1 breaker on top of the other) Single Throw / Double pole.

Sounds hard to understand but it really is not. In order to get 240 VAC, you need to use both phases available at the panel.

A panel is designed to weave the two phases together.

For example, phase A is supplied to breaker positions 1 and 2 (side by side) Phase B is supplied to breakers 3 and 4 (also side by side)

The way to get 240 VAC, you connect a double pole single throw breaker into positions 1 and 3. Red would connect to position 3, Black would connect to position 1. White connects to the "Neutral" bus bar. Usually mounted on the side of the bus bars. You will see all the white wires connected there.

Never, ever put two wires into a hole. They are designed for one conductor only. The ground (bare) goes to the grounding bus bar. Again 1 wire 1 hole.

If you are new to this, please consider using professional help. Panel work can get you killed, dead if you screw up. There is no margin for error!

Also, the correct conductor has to be used. To get the correct size for the conductors, you need to know the amperage that the appliance will need to use (load).

Once we have that, we can tell you what size cable you can use. Knowing the type of appliance can tell us whether we need to define the appliance as a "continuous use" device.

This may affect both the Amperage and conductor size required, Finally, we need to know the distance of the run, and whether you are running this to a remote building (like a garage or workshop).

Clear as mud?

Any chance you have a Federal Pacific breaker panel? They have an unusual configuration which can create this situation with a double pole breaker in the wrong position on the panel.