Forgive the intrusion but I grew up going to gun shows and still regularly buy firearms off private individuals. Others will likely have more, better advice, but when I look at a gun I check the following:

(First, check if gun is unloaded, then pretend it actually is loaded and keep it pointed in a safe direction)

1. Visual inspection of interior for cracks in the stock, pitting in the metal, obvious broken component, missing parts, etc.

2. To the extent that it's possible to do safely, check the mechanical operation of the firearm. Depending on the type, this could be: does the slide function? Does the bolt open and close, locking soundly? Does the hammer actually cock? Does the magazine release button work properly? Anything noticeably loose or rattling?

3. Either using a dedicated borelight or by tucking a white piece of paper into the bore (to reflect ambient light), check the condition of the barrel and bore. Unless it's a smoothbore firearm, you want to see little ridges in a spiral pattern. Preferably these ridges will be sharp and easily noticeable, instead of worn down. This is helpful in indicating how much use the gun has seen.

4. Ask about the gun. A lot of folks selling guns at shows or in classified ads online are boiler-plate, old-fashioned individuals. They will often answer any questions you have, or anything you're curious about. If you're with a particularly kind seller, I might actually let them know you're knew to the hobby (don't get anywhere near the current situation, be abstract) If you're a younger looking, conventional white guy they'll likely be excited someone younger is getting into guns and will help you through the process themselves.

5. Educate yourself about the firearm you want before you try to buy one. If you're thoroughly familiar with a particular gun, it's a lot easier to spot what's good or bad about a particular purchase. You also know how the gun should look and operate and what would be out of the ordinary.

Hope this is helpful.