Well, you certainly are right that this isn't normal. An outlet isn't very useful if it goes out the second you plug something in!

As for what is wrong, you don't have a whole lot of places to check. You know it works up to the GFI. That means the problem is somewhere between that and the next outlet. Check to make sure the GFI is installed properly. Buy a new outlet for the non-gfci (very cheap) and make sure it is wired properly.

Not a pro, but sounds like the second outlet is wired as a slave to the upstream GFI. I agree with bill -- buy a new outlet to replace it and install it. Make sure you turn off the breaker first and check both plugs of the outlet with a circuit tester prior to starting. To be REALLY safe, tag the breaker as OFF while you work.

I suspect highly that you do have a ground fault due to miswiring at either the GFCI or the downstream outlets. It's possible that one of your wires that's on the protected side of the GFCI is touching a wire from the non-protected side (or from another circuit).

It's far more likely that the wiring is screwed up than the receptacle is defective.

You need to open the GFCI and check the wiring. There should be connection to the load terminals for the downstream devices. The should be no connection between white and bare wires.
It could as simple as the ground wire is touching the neutral screw on any of the downstream devices.

If the ground and neutral were touching the symptom would be it would trip all the time even without anything plugged into it... Tripping with a load is most likely as I described it.

sounds like the receptacle is moving just enough when being plugged into to cause the neutral and ground to touch each other. try to reset the GFCI after you plug into the other receptacle. if you can reset, something is moving and touching.

"If the ground and neutral were touching the symptom would be it would trip all the time even without anything plugged into it... "

This depends on the age of the receptacle.

Thanks for the helpful advice folks, I opened it up and it seems that they twisted the white wires together and only have one hooked up to the gfi. Any ideas how ti fix this?

Here is a link that might be useful:

Did an electrician do that?

Remove the white connecting to the GFCI. Remove the wire nut holding the white together and untwist the wires. Throw out the short piece of white wire.

You will see the wires are bundled together in a set of black/white/green as the enter the box. Find the white wire that is bundled with the bottom black wire and connect it to the bottom screw on the side where the current white wire is. Connect the other white to the top screw.

Assuming he had the live/black wires in the right slots to begin with, that should work. If not, move the top black/white to the bottom and the bottom black/white to the top.

I can't tell from the picture if the upper set screw on the neutral side is missing. If so, you would just need a new screw.