I have a 1990 Sidekick with a 1.6L, 2 door, manual transmission and my Electronic Controll Module let the magic smoke out.  And of course it won't run, I have another Module but I am concerned there might be a short in my wiring, and if I hook it up the new one might fry.
Is there any way I can take an Ohm meter and tell which wires should have continuity and how much if any?
Any help will be greatly appreciated.

For this, you will need a wiring diagram,  several EE degrees I think, and a LOT of patience, plus to know how many ohms you need from which wires.

Carefully pop the case open and look to see if the capacitors have smoked. They're the little beer can looking items and you'll be able to tell if they're gone. Below is an attached image and detailed information from a site called kick-fix.com the original link is http://www.kick-fix.com/ECU/ECU.html From the image you can tell where and what the suspect items look like.

Hope this help, Eric

TAKE THE  BASIC TEST:
This is only a very simple test, nothing fancy.

Test 1:  (KEY ON , NO START LAMP TEST)
Car will not start, or it only runs cold well, and badly hot.  " Engine is stuck in Limp Home mode , "Fail Safe" it is called by Suzuki" Assuming all  the fuses are good and the MAIN relay is pulled in and full power is applied to the ECU. ( check it) Turn on the Ignition KEY, do NO start.  The CEL lamp  must be glowing. If the Check Engine Lamp (CEL) does not glow , the lamp is burned out (replace it, and repeat this test !!!) or the ECU is dead.

Test 2:  ( Commanding up Trouble CODES Test)
Insert the Diagnostic Jumper in to the proper plug . Key on , no start again. ( engine must not be running ! again)
Next make sure the CEL flashes code 12.  ( code 12 is normal code)  if the CEL does not flash out "FLASH , Delay, FLASH,FLASH, repeat" the ECU is dead. The only exception to that, is if ECU flashes out a chain of other error codes, telling you that something else is wrong, and that it is NOT DEAD !


Test 3:  ( NO INJECTION TEST )
You car has good  spark (or engine  runs , intemittantly but not today) AND you check the Injector with a  NOID lamp and the NOID lamp is dead. ( You discover that you have , no injection signal)
If that is true, most likely the ECU is DEAD. (check wiring).  If you have spark and no injection signal (dead NOID) , suspect a bad ECU.  99%.  (spark is ok). NOID = (guessing?,  "No Output Injector Driver"?)   ( no one knows the first usage of this word, but all mechanics know it by name). PLEASE NOTE:, the ECU is programmed to NOT provide FUEL INJECTION if there is NO SPARK.  (a safety feature)
If you don't have spark ,check the CAS. (tune up ,etc). If your ECU fails ANY of these  tests, the ECU is Dead , crazy , malfuntioning ( busted).  I would replace all the Capacitors , in it, pronto ! It is important to note, the ECU is programmed to at least start the engine and give you power to drive home , with all the sensors dead ,except the TPS and CAS!