There's little necessary to operate a code reader.  It simply spits
out the codes which indicate something about what the computer has
seen that it didn't like.  The next step after that is to decipher
what that code means.  There are a few websites that will give
information on the codes.  I simply googled "OBD-II codes" and
received a few nice hits on the first page.

Further diagnosis is where things get complicated.  Even for universal
codes (P0xxx), there can be different criteria for setting that code
on different vehicles.  To get this information, you'll need access to
a shop manual for the vehicle in question.  An accurate diagnosis will
often require knowing the conidtions that must be met for the code to
set.  The technician will, in most cases, make readings with a scan
tool, helping him conclude what is occurring with the car.  In some
cases, equally valid tests can be made with a VOM, but other cases
require using data interpreted by the vehicle's computer.

Here's an example of a diagnosis/repair I made today:
Car arrives -- check engine lamp on -- owner states only minor change
in performance.
Read codes.  P0455, P1166 (unable to properly compensate air-fuel
ratio, bank 1), P1167 (unable to properly compensate air-fuel ratio,
bank 2)
Read freeze frame data (conditions present when code set).  Freeze
frame data indicate conditions when P0455 set.  Notice short term and
long term fuel trims* very high positive both banks.
Clear codes.  Check fuel cap.  Run evaporative self-test.  Test
passes.  Conclude fuel cap was left loose or off, causing P0455, then
later installed correctly.
Check current data and verify both fuel trim values still high.
Check air flow readings.  Find readings low compared to known good
readings.
Check PCV valve, brake booster, piping/hose between intake and purge
valve, fuel pressure regulator hose for vacuum leaks.  All okay.  Find
idle actuator gasket out of position and leaking air into engine.
Reposition gasket and recheck data.  Fuel trims now normal.

*Fuel trim represents the adjustment the computer makes to the
injector on-time in comparison to what the computer would expect based
on air flow readings, throttle position, engine rpm, and other
factors.  Positive numbers mean the injectors are being kept on
longer.  Negative numbers mean the injector on-time is being adjusted
smaller.

This diagnosis is a good example of the ability to use a scan tool to
help in diagnosis.  A do-it-yourselfer would never have been able to
run the evap. test.  And although a do-it-yourselfer wouild have been
able to check all the things I checked for P1166 and P1167, using the
scan tool provides for near-instant feedback of results.  I can check
for leaks by pinching off a vacuum hose and watching the fuel trim
numbers.  If pinching the hose stops a leak, the fuel trim numbers
will come down.

This diagnosis is also a good example of how to use logic.  Oxygen
sensors are the single most frequent cause of P1166 and P1167 codes on
the vehicle on which I was working.  In this case, however, they
weren't the primary suspect.  Why?  Because I have a near-identical
problem on both banks, and it would be highly unlikely for the sensors
on both banks to fail simultaneously and in the same manner.  Also
note that it's improper to conclude that because the air flow readings
are low that the air flow sensor is defective.  In this case, the
readings were low but accurate.  The high fuel trim values were due to
the computer needing to compensate for the additional are leaking into
the engine that wasn't being measured by the sensor.  Low fuel
pressure could have also been the culprit, but that would have been
much more likely to cause a significant power loss at high load/high
rpm and result in a significant complaint about performance.  If the
air flow readings been correct, checking the fuel pressure would have
probably been the next step.

Can't boil it down to that level of simplicity.  Too many computers and too
many inter-related systems today.  Just like that 57 Chevy, it requires a
certain level of familiarity and experience with the vehicles to wind your
way through such things as multiple codes.  The 57 was capable of showing
you multiple and confusing symptoms - today's cars just raise that to a new
level.