Your car should be well maintained, at the very least. Make sure to keep up with oil/fluid changes, check your brakes, etc. Remember that the timing belt is also very important. It should always be changed within the recommended service period. In autocross, your engine will experience high revs and they could be more than an old belt can take. Make sure to check all of your fluid levels, and tires for sufficient tread and pressure before race day.
This one’s important… clean out your car! Remove absolutely everything that you won’t need for race day, and make sure you take everything that you bring with you out before taking your car for tech. As part of tech inspection, officials will check the interior of the car to ensure nothing is loose in the vehicle. Loose items can end up flying around the car while you’re out on the course, which is dangerous. You may also want to wash your car and the engine compartment if you haven’t recently.
Just before you get to the autocross site, it may be a good idea to pump up your tires to 42 psi or so. It sounds high, but you will need the extra air to prevent rolling over onto the sidewalls during spirited driving and hard cornering. If you’re a novice, just ask fellow drivers at the site how much air you’re likely to need. Then just let out enough air until you reach that pressure. It’s much easier to let air out than it is to pump it in!