There's not really anything substantial you can do as a field inspection to look for any major problems other than audible warning signs.
It's worth it to take it to your local (trusted) mechanic and spend $100-200 for a full inspection of your car.
This may be too late for you, but you should really take any car you want to buy to an independent mechanic for an inspection before you buy it. View the cost of the inspection like a home inspection when buying a home. The inspection fee is a small price to pay to save you from possibly thousands of dollars in expenses down the road.
Also, while carfax isn't perfect, it's worth it to run a check on the car to make sure it's not a lemon or had any major damage.
For the 'rough' check I would say, use all your senses.
- Listen for strange noises.
- Look over the whole car, see if the body panels all line up, if all the openings work well (hood/doors opens/closes properly)
- Feel the body panels see if you notice any warping that would result from body putty.
- Smell the interior, sniff the exhaust for the smell of burning oil etc.
- Taste - well, if you get a bad taste in your mouth from looking at the car, it's probably not good for you :-)
Most of all, ask questions! Car sellers are required by law to answer all questions truthfully, but if you don't ask they won't offer up any negative information.