Car starts, then won't, then does--- fuel pump?

I'm new here, but I appreciate all the help others have gotten. I've worked on many cars before, but this one is stumping me.

Lately, my 2001 Ford Taurus will not start. It cranks, but is not getting fuel. When the key is turned, the fuel pump does not prime (no humming), and won't start. Then after just trying a few times, it will humm, and start right up.

I'm trying to narrow down what is causing this, before just replacing the fuel pump, to avoid the cost if possible.

I checked out the relays, and they are operating correctly, but was wondering if they would do this intermittently?

Any advice is greatly appreciated, it's cold up in Wisconsin!