The "Check Gauges" light means look at the dash gauges to see which one is reporting something wrong. They put that light on most cars now because gauges that move slowly don't get our attention like the older "idiot lights" used to do. Too many of us don't look at those gauges until it's too late and the problem is already occurring. Be aware the oil pressure is going to be 0 psi when the engine is not running. Electrical system voltage will be low when the generator is not working, (engine is not running). Both of those conditions will trigger the "Check Gauges" light, which is to be expected. The engine not running is not a normal condition, so that warning light can be ignored. It is only relevant when the engine is running.
By "turn on", do you mean the starter will not crank the engine, or it cranks fine but doesn't start and run? If the starter doesn't spin the engine, the battery could have failed, but given your other observations, a better suspect is the generator has failed. You can verify that by charging the battery for a couple of hours, on a slow rate, with a small home battery charger. If the engine cranks normally and runs after that, at most it will run for about an hour until the battery runs down again. That will give you time to drive to a repair shop to have the charging system tested. You'll be able to drive for a much shorter period of time if you use a lot of electrical stuff like the heater fan and head lights.
The "Check Engine" light could be related to the first problem, but in this case, the Engine Computer will have stored a diagnostic fault code. Those codes never say to replace a part or that one is bad. They only indicate the circuit or system that has a problem, or the unacceptable operating condition. Those codes tell the mechanic where to start looking to solve a problem. The people at many auto parts stores will read those codes for you for free. Be sure to write down the exact code number(s) if possible, not just the written description.
If the generator is not working, the resulting low system voltage can cause the many computers to do weird things. The Engine Computer could be setting a fault code telling about the failed generator, or a code related to something that stopped working because of the low voltage, or it could be a totally unrelated problem. We can figure that out once we know the code number.
Be aware too that starting with the redesigned generators for the '87 model year, GM has had a huge problem with repeat failures. I can share the gory details if you want me to, but to reduce the high number of repeat failures, if you need to replace the generator, replace the battery too, unless it is less than about two years old. The old battery will work fine in an '86 or older model.
Sunday, July 2nd, 2017 AT 11:34 PM