The first thing you should do is to do research on the car you want. You should read consumer reports, reviews, reliability ratings and data on gas consumption of the makes and models of the cars that you are interested in.
Compare them with each other and try to determine what the maintenance costs are of the cars that you are interested in. You should narrow your search down to one or two models. There are two reasons for this: If you don't know what you want, you won't be happy with it. You may find out too late that it is not exactly what you wanted or that you can't afford the maintenance on it. Also, when you test-drive a few cars of the same model, it will be much easier for you to compare the condition of each car and to pick the best one.
Where should you start looking for a car? The internet has many websites that list cars or car dealers. You could start by looking at these and/or eBay's used car section.
When you find your car, you should check its history records. This will enable you to eliminate the vehicles from your list that have potential problems. These include cars that have been restored after serious accidents, had their odometers turned back, cars that have been flooded or ex-rentals. Don't buy a car based on what you've been told by the dealer because it may not be the truth. Ask for the car's papers and check the history yourself. Then use a list to check for various things in the car. Finally, get the mechanic you trust to come and do an inspection. NEVER give a deposit before you and your mechanic inspected the car.
Then you should be careful when doing the paperwork. Make sure that the person who signs the Bill of Sale is the owner of the car. Also do a check with the Vehicle Registration Authorities that the car has not been stolen.
If buying from a dealer, read all the papers especially the fine print very carefully. If it's a "Certified" used car, you should check which items were checked off and approved because sometimes the car have a history of an accident, or come with a poorly maintained engine and still get Certified. The dealer might make promises to you. Make sure you get these in writing. If there is a warranty on the car, make sure it can be transferred to you.
When you are happy with the car, you could start negotiating a price. Always try to get a discount. Don't sign anything or give a deposit before you have agreed on the price and you have it in writing. Once all the paperwork is prepared, read it through one more time before you sign it to check that everything is as agreed.
Now you have a car that you are happy with. Congratulations!