The radio and cigarette lighter doesn't work any more and I'm assuming it is a fuse problem. Where do I locate the fuse box on a 1993 Toyota Corolla? And is this something that I can fix on my own?

Car manual has exact location of all fuse boxes and description of circuit covered.

Please check the radio fuse.

Check these two fuses. Fuse number 31 in fuse box location 4, if cigarette lighter, radio and clock. Else fuse number 5 in fuse box location 2, if only radio and clock and not cigarette lighter.

I found this useful diagram at troubleshoot vehicle.com. It answers your question. Fuse box is located below steering column.
Driver Side Fuse Panel
(1993-1996 Toyota Corolla) December 21, 2013 Updated: December 21, 2013 Written by: Abraham Torres-Arredondo In the table below you'll find the fuse location and description of the fuses of the driver side under-dash fuse box on 1.6L and 1.8L 1993-1996 Toyota Corolla.
The fuse box is located on the driver side of the dash, below the steering column.
1993-1996 Toyota Corolla Driver Side Fuse Box Fuse Fuse Name Amps 1 Cigarette Lighter, Digital Clock, A/C System, A/T Shift Lock, Alarm, SRS Airbags 20 2 Tail, Parking & License Lights, Instrument Panel and Daytime Running Lights 15 3 Fuel Injection, Cooling Fans, Battery Discharge Light 10 4 Cruise Control System, Daytime Running Lights 10 5 Fuel Injection System 7.5 6 Stop Lights, Cruise Control, Shift Lock System, Anti-Lock Brake (ABS) System 15 7 Turn Signals, Emergency Flashers 10 8 Starting System, Anti-Lock Brake (ABS) System, Shift Lock System 15 9 Gauges, Reminder Lights, Back-up Lights, A/C System, Daytime Running Lights 10 10 Front and Rear Wiper/Washer System 20

On the driver -side kick panel are the fuses located. Take out the panel,check 7.5 dome,15 amp cigarett lighter and 15amp tail. If it pop off again,you have a short ckt.

you may have a blowen fuse, i know was working on the a radio today and blew the fuse, located under the dash board

The fuse you are looking for might be in the fuse box under the hood. I think on or about page 203 of the owners manual it will show the locations of the fuse boxes. The one under the dash is a little tricky. Just figure out what by looking at the pattern of the fuse lay out. You are looking for a 15 amp(blue) fuse I believe.

After you fix that I would recomend using the power outlet. I believe there are 2. One just in front of the shifter and the other in the center armrest storage box.

Also, for radio, cigarette, and others: behind the "storage only" box left of steering wheel. Pull that door open, up and out.

The radio, lighter and clock are controlled by 2 fuses:
The 15 Amp CIG and the 15 amp DOME, under the dash, driver side, for a USA car.

The fuse you are looking for is under the hood. There's a fuse block out there for some 'other' things. It should be on the drivers side, black plastic looking box with a hinged lid (press here to open). Be sure that you don't have a coin or something stuck in the cig lighter before you replace the fuse.