trigga, trigga, trigga - you still didn't tell us if you have a multimeter or test light. If you don't have one & switching relays doesn't help then you'll need to get one.
If you remove the fuel pump relay the terminals on the relay should be numbered 85, 86, 87, 88? & 30. Put the meter leads where terminals 85(pos)& 86(neg) would plug in. When the key is turned to run you should have 12 volts for 2-3 seconds. It sounds like this should check good because you can hear the relay clicking. Now turn the key to the start position for a couple seconds & you should have 12 volts while the key is in that position. This will confirm the ignition switch & PCM are energizing the relay. Now move the lead from terminal 85 to 87. You should have a constant 12 volts with the key in the run position. Let us know what you find.
1999 Pontiac Bonneville SSEi (son'*)
1997 Chevy Lumina (daughter'*)
2006 Chevy Trailblazer (wife'*)
2006 Chevy Monte Carlo SS (Mine! -Zoom Zoom Zoom)