Simple way to answer this question:
Has the car been properly maintained?
If so, start driving and don't worry.
If you aren't sure, look at your maintenance records and your car's service schedule. Take care of whatever isn't up-to-date.
If you don't have maintenance records, you're very lucky to get this far. Start shopping for a new car.
My quick checklist before taking a trip:
- Check all fluids (brake, power steering, ATX, engine coolant, engine oil). If any are near a scheduled change, take care of it now.
- Check all tires (tire pressure, tread depth, etc.). Don't forget the spare.
- Check over belts, hoses and other "consumables" (like wiper blades) to make sure none are near the end of their life.
- Has the car done anything unusual lately? If so, get it checked before leaving.
I tend to buy used cars and keep them along time, so careful maintenance is key to saving money and avoiding breakdown. Case in point: I'm currently using my '86 Mustang as a daily driver at 207K miles. The car has always been reliable, but I've also been nearly fanatical about maintaining it. The car shows every sign of giving many more years of good service.