In what order do I connect a landline phone to a caller ID and to a fax machine from the wall jack so that all three will work properly?

Assuming one line: Wall Jack > Caller ID Unit > Fax > Phone.

It also depends on what jacks the Caller ID Unit has. Might be able to plug both phone and fax into it. Most faxes have a jack to plug a phone into. Might need a 3-way splitter if Caller ID and Fax don't have jacks to plug anything into.