Oil
Changing the engine oil every 3,000 miles or so is standard car care, unless you are using synthetic oil.
Air Filter
Your manual will come with a recommendation. You can go with that or have the air filter checked with every other oil change.
Tires
Properly inflated tires increase fuel efficiency, reduce wear, and increase safety. This should be checked every other week.
Fluid Checks
This is a highly neglected form of car care. The fluids for power steering, transmission, brakes, and radiator coolant all play large roles in the healthy functioning of your car.
Engine Belts
The timing belt in your engine should not have major cracks or miss segments or have broken teeth. Minor cracks on the glossy surface are all right, but deeper cracks will require a replacement. There are a few things to beware of:
Chassis Lube
Chassis lube used to be a standard part of car care packages sold at oil/lube shops. However, with the more common use of higher grade, soft, pliable rubber in various sections of your car, your car doesn't need it, or at least not as frequently. There is no downside to applying chassis lube, but be aware that many lube shops will no longer apply chassis lube even if it is part of their “premium” package.
Flushes
Not all flushes are necessarily good for your car, and some may damage the components. Automakers have stated that you do not need any maintenance beyond what is listed in the owner's manual. Check the details on your particular vehicle to be sure.
External Car Care
A car should be washed once a month, though once a week is probably overkill. It is important to remove things like bugs, bird droppings, or limestone drippings as soon as possible because their corrosive nature can leave permanent damage in the paint. Consistent waxing will help shield your car from harsh environmental factors.
Rustproofing
Areas along the coast with high humidity and high salt content should probably rustproof the underside of the car. The metal components underneath your car, such as the brake system, can be corroded within a matter of years if rustproofing is not applied.