This just started about 3 weeks ago.

The first time I turn the ignition key, I get no ignition - just like there is no fuel. Engine cranks fine.
The second time I turn the key, the engine fires up immediately - every time. What could this be?

 

This could be a few things !!...just turn the key to on and listen for the fuel pump running !!...after 2 seconds it will shut off if you don’t try starting the car...if you don’t hear the fuel pump...that’s bad !!...if you DO hear it running see if the car starts normally...you can start there !!

 

I forget what it's called, but there is something that prevents pressure from bleeding back into the tank when the car is off that can go bad. When all the pressure bleeds out of the lines, that makes it a little hard to start next time until it reprimes. Mine does it every now and then, so I just let the fuel pump prime for a moment before I turn the key the rest of the way.

 

I've noticed that mine has been more stubborn to start recently - I am discussing this with ECS and the first question they asked is "How old is your battery?"

 

You may have a problem with fuel pressure bleeding off, there is a check valve that prevents this. I normally perform a second key off, key on before turning the key to start, it seems to fire just a little faster, or it could be my OCD lol

 

I have never been able to hear my fuel pump in 20 years (maybe I am deaf?), but if the fuel pump were dead, how could I drive around all day once the car starts?

 

Your problem indication does sound like a fuel feed issue but to be sure, pull the codes and post up what codes you have; If you have no codes that could relate to the problem you are having, then the next step would be to test the fuel feed system.

Below troubleshooting may be helpful; you can get a inexpensive fuel pressure gage testing kit from the local "freight" tool store...
Fuel System Description
Fuel Pump Electrical Circuit
When the ignition switch is in the ON position before engaging the starter, the PCM energizes the fuel pump relay for 2 seconds, causing the fuel pump to pressurize the fuel system. If the PCM does not receive the ignition, reference pulses with the engine cranking or running within 2 seconds, the PCM shuts OFF the fuel pump relay, causing the fuel pump to stop.
Starting Mode
With the ignition switch in the ON.RUN position, before engaging the starter, the PCM energizes the fuel pump relay for 2 seconds allowing the fuel pump to build up pressure. The PCM first checks speed density, then switches to the mass air flow (MAF) sensor. The PCM also uses the engine coolant temperature (ECT), throttle position (TP), and manifold absolute pressure (MAP) sensors to determine the proper air/fuel ratio for starting. The PCM controls the amount of fuel delivered in the starting mode by changing the pulse width, or the On time, of the injectors.

Couple of tests you can do to see if you are losing the static fuel pressure with the fuel pump OFF and not activated.
1. Loosen the fuel filler cap in order to relieve the fuel tank vapor pressure.
2. Remove the left fuel rail cover.
3. Connect a fuel pressure gauge to the fuel pressure connection. Wrap a shop towel around the fitting while connecting the gauge in order to avoid spillage.
With the ignition switch, cycle the key from OFF to ON/RUN position (NOT START) to pressurize the fuel feed system. Repeat this step 5 times to build fuel pressure. Record fuel fuel pressure on gauge which should be around (55-62 psi). Important: The fuel pressure may vary slightly when the fuel pump stops running. After the fuel pump stops running, the fuel pressure should stabilize and remain constant.
Monitor the fuel pressure gage for 1 minute.
If the fuel pressure drops more than (5 psi) then you have a leak in the fuel feed system.
If during this test, there is NO fuel pressure, then you will have to troubleshoot the electrical side if the fuel pump command circuit.

If you perform the above test, post up your results,
Cheers,
Goose