One of the biggest warnings you should never ignore is how hot your car is running. Never, ever, under any circumstances drive your vehicle when it is overheating or about to overheat. The engine inside most newer cars is made of aluminum. Heat and aluminum do not get along very well. If you overheat an aluminum engine the chances are you will warp the head and your head gasket will fail, or worse. This can mean thousands of dollars in repairs. If your vehicle begins to overheat, turn the heater inside your car on full blast and pull over as soon as you can. The heater in your car pulls heat from the engine to warm the cabin, so it may buy you a few precious seconds. If you notice your car is starting to run hot, get the problem investigated before you end up pulling off on the side of the road with steam billowing out from under your hood. A $12 thermostat can cause huge problems and make a mess of your engine if you ignore when things get hot.
In my opinion the biggest warning sign your car can present is low oil pressure. This is usually displayed as a little red light with a picture of an oil can. If this light comes on while you're driving it means the engine isn't producing enough pressure to properly lubricate it's internal parts. This is the equivalent of having a heart attack. Without engine oil, or oil pressure, the engine will definitely fail and leave you stranded. When the light comes on you should do everything safely in your control to turn that engine off as quickly as possible. This is where seconds matter as the engine can't survive without oil. Whether it's leaked out, or burned off completely, an oil pressure warning light should never be ignored. A good rule of thumb is to check your oil ever time you get gas, or ask the attendant to check it for you. It's cheap insurance against a catastrophic failure.
My focus here is not to make you feel bad for not maintaining your car. I understand money can be tight and if something isn't broken then don't pump money into it. The exceptions I've listed above do warrant your attention though to make sure you're safe and don't find yourself on a walking tour of our nation's many interstates.