If your car is not starting due to the alarm, it is because the alarm is still active. Often the likely suspect for this problem is the door latch (or more than one) The door may physically unlock with the key or remote, but will not recognize the "door open" position when opened, and the alarm will re-arm within a minute.
You can test the door latches- unlock all the doors, turn the key to "on" in the ignition, and turn the dome light switch to "door open". Try opening one door at a time, and see if the dome light comes on- if it does not, that latch is electrically faulty. The light should also fade out as soon as the door is closed, if the key is on.
If the latches are OK, you may want to check a couple other switches. Arm the alarm with the remote, and try pushing in on the radio on both sides- there is a microswitch that detects if the radio is not seated fully (!). The microswitch in the hood latch has a wiring harness that leads over to the passenger side front of the radiator area- follow the wiring over there and unplug it, and try arming the alarm again. If it arms, then that switch is faulty.
There is also a switch in the trunk- when you open the latch, the courtesy light in the trunk should come on, and go off when the trunk latch is closed (you can simulate this by simply closing the latch with a screwdriver shaft rather than actually closing the trunk. Try arming the alarm again, and close the latch.