There is one outlet with the GFCI and several other outlets without. The GFCI outlet works fine, while anything plugged into the other outlets causes the GFCI to trip.
How should I detect and fix the problem?
A GFCI trips when the current coming in through the hot and out the neutral are not equal. The fact that your GFCI trips whenever something is plugged into a LOAD-side outlet, but not when plugged into the GFCI itself, suggests to me that there's a neutral-to-ground-fault (neutral wire touching ground) somewhere on the LOAD side.
This is the way I would diagnose your problem:
The above is not a one-hour job; it will probably take you the better part of a day. It will take longer if you don't already know how the connections between the outlets are laid out.
If you're not comfortable with any of this, hire a licensed electrician.
It sounds like the outlets are daisy chained to the GFCI. The fact that the GFCI is tripping is a good thing, because you most likely have a wiring problem downstream from the GFCI.
Electricity is the one thing I am wary of doing myself unless I am 100% confident in the task; so my best answer to "howto diagnose" is to call a reputable electrician.
You may have 2 or more hot circuits with a shared 1 neutral (white) wire. See if the outlets that trip the GFCI are on a separate circuit breaker.