Check your tyre pressure before every ride, your tyre will have the recommend pressure printed on it.
Lubricate your chain. This should be done every time your bike has been cleaned or been in the wet. At a minimum you should do it once a month. A light or teflon based oil is recommended as it collects less dirt and grime.
Riding, do not use your bike for riding styles that it was not designed for. For example, riding a racer on bush tracks (see Choosing The Right Bike).
Noise, be aware of the noises that your bike is making. Ugly noises are generally a good warning sign that your bike is not running exactly as it should be.
Similar to your car, your bike should also have a regular service. Gear and brake cables do stretch and will need to be adjusted from time to time, especially if your bike is only a few months old. Your wheels may get slightly buckled which should also be fixed during a service. Your local bike shop will be able to service your bike for you. A service could cost anywhere between $30 and $150 depending on what needs to be done and what parts need to be replaced (see Bike Shop Directory).