Pressure
Check tire pressure monthly with a real gauge, with the tires cold (before driving). The recommended pressure is on a sticker in the driver door jamb, not on the side of the tire — the side of the tire shows the maximum pressure the tire can safely hold, not the right pressure for your car.
Tread
Insert a quarter into the tread with Washington's head pointing down. If the top of his head is fully visible, the tire is at or below 4/32 of an inch and needs replacement soon. A penny works for the absolute minimum legal tread (2/32) but waiting that long is dangerous in rain.
Rotation
Rotate tires every 5,000 to 8,000 miles to even out wear. Front tires usually wear faster than rear tires. Rotation is a cheap service and many tire shops include it free with the original purchase.
When to replace
Replace tires that are six years old or more regardless of tread, replace any tire with a sidewall bulge, and replace any tire with a puncture in the sidewall (the side, not the surface that contacts the road). Sidewall damage cannot be safely repaired.