I strongly believe that finding the right car for you is a highly individual choice. No one can tell you what's right for you, you need to figure it out for yourself. That being said, there's plenty of useful advice people can offer and information to be found to help point you in the right direction.
Figure out what you need. Do you carpool? Need to haul your kids and all their stuff? Do you need to be able to drive offroad? Is gas milage important? Set a budget. Figure out what your absolute must have features are.
Figure out what you like. Start with body types, do you prefer a sedan or a hatchback? Do you like to drive fast? Do you like a smooth comfortable ride? Are there certain makes you find yourself more attracted too?
Do research. Start reading about the cars that fit your criteria. I often find wikipedia is a good starting point to give you an overview of the car. Look for reviews online by professionals. Read forums dedicated to the car, try and get as much info as you can written by people who have actually owned the car.
Narrow it down. Pick a few, maybe 3 or 4 to focus on.
Check used car listings. Start scouring used car listing sites like ebay and craigslist. Check at least every few days. Figure out what the market is like: what's an average price? Are they common or rare? Are there examples available in good condition? Do they sell quickly or are they lingering for months?
Visual Inspection. There are plenty of guides out there on what to look for when checking out a used car, read them. Eric the Car Guy has a great video guide on YouTube. Doing a visual inspection can help you weed out obviously bad choices.
Ask questions. Don't be afraid to ask anything. Common questions I like to ask are how long they've had the car, why they are selling it and if they have service records. Try to get a feel for the person, how well they know the car and how they treated their car. Don't take anything for granted.
Test drive. Test drive as many cars as you can. If the car is available at dealers that can be a convenient way to test drive a car, just don't get bullied into buying something. If they start pressuring you tell them there are other cars you're considering before you can make your decision.
Be patient. Don't feel pressured to buy anything, there are always other cars and other deals out there. Always be willing to walk away. If the seller is making a big deal about how they have so much interest in the car and it might be gone later, they're probably talking out of their ass. And even if they aren't you don't want to rush into making a decision anyway. Be prepared to spend months looking for the right car.
Take it to a mechanic. You've found an example you like, the price is competitive, and the seller doesn't seem like they've abused the car, now have a professional inspect it. Buying a car is a big decision and you want to know exactly what you're getting yourself into. This can get a bit expensive, but it's better to spend a bit of money on an inspection now than end up having to pay thousands of dollars later. This applies to buying used cars from dealers too. Most dealers are going to do the minimum to get the car in a condition to be sold legally, this doesn't mean they've done anything to address any potential time bombs that are waiting to go off after you've got the car home.
Trust your gut. If something doesn't feel right, or you're just not sure about it don't do it.
Knowledge is power. The more you know about the car you want and what the market is like, the more likely you are to make a good decision.
Feel free to add your advice in the comments.
A note on the image I used for this post, it was the first result returned when I searched for "used car" on wikimedia commons, I find that amusing.