You might have a bad alarm switch in the driver door handle. Unlock the car from the passenger side or the trunk then see if it's back to how it was before using the remote. If that does it, don't lock the driver's door by key until you get the switches checked.
In the interim, locate the alarm control unit. I think it's under the almost vertical plastic panel in the passenger footwell. Look for a box maked 'Becker.' It'll have a couple of ~12 pin connectors. Detach the connectors to disable the alarm. The door locks and remote keyless entry should function normally with the alarm disconnected.
There's lots of info in the archives on getting keyless remote to work with the factory alarm system.