Check HVAC system filters. Some filters are reusable, while others are disposable and must be replaced. Clean or replace filters when they get dirty, which might be anywhere from 30 days to a few months.
Look for leaks around toilets and sinks. Any sign of water where it shouldn't be is reason to investigate further. Even a tiny leak could cost hundreds of dollars each year, as illustrated by this chart from the City of New York.
Inspect grout and caulking. Touch up any voids or cracks in tubs and showers. This keeps them looking nice while helping to avoid seeping water damage.
Check kitchen vent hood filter. Clean or replace if needed. Consider looking at the vent hood filter more often if you have an avid chef in the household.
Test smoke and carbon-monoxide detectors. In addition to a monthly test, Consumer Reports explains why you should change batteries every six months. Make a habit of changing the batteries when you change your clocks during Daylight Savings Time.
Stroll around outside. You're probably outside your home on a regular basis, but how often do you really look it over? Walk around the house and any outbuildings, looking for problems with the foundation, vents, gutters and drainpipes.