To retrieve the stored codes off the OBD on the car - you need to short the diagnostic pins on the port under the hood. They make a special short pin to do this, or you can fashion your own out of some wiring or even a paperclip.
Here is a link on how to do it - http://www.troublecodes.net/Toyota/
- basically after the pins E1 and TE1 are shorted, the ECM will fall into a diagnostic mode and \"flash\" the trouble codes through the check engine lamp. They do make OBD-I code readers, but they won\'t tell you anything more than what this \"free\" method with do. Different circumstance with OBD-II systems (1996+ vehicles) as the information is too complex to be flashed by the onboard system.
As for the trouble codes, they are more or less universal, but just in case - these are the ones listed for Toyota specifically:
21 - Oxygen Sensor Fault
25 - Air-Fuel Ratio Lean
26 - Air-Fuel Ratio Rich
43 - Starter Signal Fault
For the transaxle, I agree with the mechanics, likely to be an issue with the shift solenoids or the lockup converter. First thing I would try is to do a full transaxle drain and refill, pan drop and cleaning, replace transaxle filter, and a good lookover the connections to the solenoids. Transaxle flushes are only OK if they use the machine that connects directly to the pump - the ones that require the transaxle pan to be removed. The cheaper machines that connect to the cooler lines can cause deposits and grit to be circulated through the transaxle, potentially leading to more problems/make things worse.
Not sure when those codes were stored, but this seems to point at electrical gremlins. If there was any recent electrical work done to the car - new radio, alarm system, aftermarket electronics, etc. - I\'d look into those installation to make sure that no other wiring was damaged in the process.
2002 Corolla S, Silverstream, 1.8L 1ZZ-FE (VVT-i)
2003 Matrix XRS, Cosmic Blue, 1.8L 2ZZ-GE (VVTL-i) - RIP
2009 Matrix XRS, Blue Streak, 2.4L 2AZ-FE (VVT-i)