Tip 1 - Check your dipstick and look at the level of course, but also look at the color and give it a sniff. Does the color look light and clear or dark or (worse) chocolate? What about the smell - does it have a burnt or otherwise odd smell? If it's weird in any way, find out why or get someone who knows about such things to check it out.
Tip 2 - Give your tires a look. Not so much as the sidewalls but give the tread a good look as well. Is it even? Can you see the wear bars starting to shine through? Are the tread blocks even or do they look sort of higher on one side than the other? Is one side of the tread pattern worn more than the other? Anything other than smooth, even wear indicates an unseen problem that will wear out your tires long before their intended life. It could be a simple alignment or it could be worn ball joints or loose steering components or even possibly worn suspension parts. Worn tires are not the problem, they are a symptom of a bigger problem.
Tip 3 - Give your electrical system a quick performance test. Start the engine and at a fast idle, start turning on various systems starting with a turn signal, then turn on the headlights, followed by the heater blower, then the windshield wipers, then the brake lights, etc.. Basically you want to slowly build up the load on your alternator/generator from very low (idling) to full load (everything turned on). As you are adding more and more systems, watch your turn signal blinking speed. Is it slowing down a lot or possibly even failing to blink at all with the additional load? With the full load applied to the system, raise the speed of the engine and see how much the blinking speed changes. If everything seems to go to "full speed or full brightness" with the motor revved up then your battery may be weak even though your charging system is working well. If nothing changes when you rev it, your charging system is not up to the load and could fail you at an inopportune time.
Tip 4 - Test your choke system (carbureted engines) before starting your engine. If your engine has an automatic choke, look to see that it is closing off the airhorn on top of the carburetor. Even on a mild day, the choke should be fully set when the engine is cold. On some engines (GM), you may need to depress the accelerator to release the choke to the closed position but the point is to ensure that the choke is actually closed before turning the key. If not, you may be having trouble with cold starts. If your choke is electric, it should be closed without having to do anything special. The electric circuit opens the choke. It does not close it.