As Bob says, formal servicing should always follow the manufacturer's schedule - Make sure to take into account any variations suggested for your location/climate/usage pattern. This should include routine things like oil and filter changes, as well as more major, but less frequent things such as timing belts.
In terms of more general checks and inspections, you should check all the lights, wipers, fluid levels and tyres every week or two (depending on how much you use the car - one suggestion I've heard a few times is "every time you fill the fuel tank" as a rough guide). You should also check door, bonnet (hood), and boot (trunk)/tailgate hinges for squeaking or debris, as improper lubrication could cause the hinges to seize leading to body/paint damage.
Wipers should be replaced every 6 months or so, and tyres when they get down to around 3mm (1/8") of tread. Bulbs should be changed as soon as possible when they have failed - in many countries it is illegal to drive with any non-functioning lights on the car*. Any minor leaks found during these routine checks should be dealt with as soon as is sensibly possible, before they develop into major leaks! The exception to this is any leaks, however minor, in the brake hydraulics - if this occurs, unless you know what you are doing, stop using the car immediately and seek experienced help.
I usually check brake pads slightly less frequently than this - about once a month or so.
*Some countries also require you to carry a full set of spare bulbs, and this is sensible regardless of where you are...