Monthly, you will need to replace your air filter. You always want a clean filter. Your system is likely to last much longer if you do this. If there is an arrow on your air filter, make sure it is facing the direction of the air flow.

Yearly, you need to have a certified HVAC technician perform a check on your furnace. The technician will do things such as lubricate moving parts, change filters, check blower components and inspect the heat exchanger for cracks. This step is extremely important. In the event that you have a cracked heat exchanger, carbon monoxide could be escaping into your home. Carbon monoxide is known as the silent killer due to the fact that it is colorless and odorless. 

Many heating and air companies offer maintenance agreements. Typically these agreements come with 2 visits per year. One visit on your air conditioner and one visit on your furnace. At these visits they will perform a 15 point check. Companies such as AccuTemp even offer discounts on repairs for their maintenance agreement customers as well as preferential treatment in scheduling no heat and no cool visits.