2006 Mazda 5 won't start, no clicking noise when I turn key in ignition?

How do I test the ignition switch? Unsure if the security light is on or not.

You can test it manually by removing it and testing for continuity in each position. Otherwise, you can backprobe the connector to test the values in each position. The security light would indicate that you are locked out.

Should I just buy a DMM then? Can it do all the work that a test light can do and more? How do I backprobe the connector exactly?

Both is best. You backprobe by sliding pins (or paperclips) into the back of the connector while it is still connected to the component. If you place them in the same hole as the wire you wish to probe, you can make contact with the terminal.