Tires are going to save you money in fuel cost, when you properly maintain them. I would recommend that you check the air pressure a least every couple of days. You do not need to fill them with air every time, just check the pressure in them. This can also be a good way to find if a tire is losing air, that way you can get it repaired.
The correct tire pressure for your car is located in the owners manual or on the driver door label. They can also be other places has well, best to check the owners manual first. A recommend setting will be given as a psi reading anywhere from 28psi to 60psi. A good pressure to keep them at if you can not find the pressure label is 32psi in all four tires. Other things to look out for with your tires:
Vibration in the steering wheel: If you are driving and the steering wheel seems to vibrate in your hands, you could have a out of balance wheel or a bent wheel. If you have a vibration, and it is bugging you, get it checked out. A flat stop in the tire can also cause a vibration. If you have not driven your car for a few months, this might be the case. Try setting the tire pressure to 55psi, drive around for awhile. Reset tires to original psi, and see if the vibration is gone.
Strange noise while driving (humming, chirp): This can be caused by a flat spot or a unevenly worn tire. If you have done the above action for the flat spot and the noise is still present, the tire maybe worn unevenly. Turn the wheel of the car so you can inspect the tire.
Steering alignment: This can cause poor gas mileage and premature tire wear. If the steering wheel is off center, then you have a alignment issue. You could have hit something that caused a part to bend, giving you the off center steering wheel. It might also be that time or mileage for your cars check. Always check your owners manual for suggest time intervals. I recommend that anytime you get four new tire you get an alignment. This will help protect your new investment. You might also want to look in to nitrogen filling your tires. Oxygen will leak out of tires faster because of the smaller molecules. The larger nitrogen molecule will not dissipate as fast. You can always top off the tire with normal air, the mix will not hurt.
The tires are your only contact to the road surface. If the tire is under inflated and the steering wheel is off center, who may have to work a bit harder to keep the car straight. This is where proper maintenance and care can save you money.