The power outlet in 2003 Dodge neon keeps blowing the fuse. The power outlet has been replaced per a previous suggestion and still keeps blowing.

The light that says "FUSE" is refering to the IOD (Ignition Off Draw) fuse. It is located in the engine compartment Power Distribution Center (PDC). For more information, This subject is covered in the vehicle Owner's Manual.

there could be a short somewhere in the wires

You may have to replace the fan.If it is burned up the fan will not turn,thus blowing fuses.

Are you using the same power feed from the old radio? If so, your new radio may require a higher current fuse. Check the manufacturers information the came with the custom radio and make certain the current (amps) required is the same.
If the new radio has a higher power amplifier for increased audio, it will blow a low rated fuse.
On the other hand, it sounds as though you may have pinched the power feed wire in the radio bracket, trim cover or have it resting agaist a sharp bracket. So when you put the car in drive, the dash flexes enough to cause the short.
Does it blow the fuse when you shift to reverse?
let me know.
Regards,

The wattage of the new headlight bulbs may be to great and exceeding the amperage limit of the fuse.

Sounds like a short in the system. you will need to inspect the wiring and posably, bench test the starter. Check the cables to the battery. Make sure the wires and the battery are in good shape.

look in your manual. it's poorly marked under the hood, but between using the manual and the fuse box cover under the hood it's easy to figure out.

i had to do that with my 2002 neon.

fuse should be located the end of the dash drivers side