Important: Before riding any bike, always check the quick release mechanisms on the wheels. Make sure they are tight and adjusted properly. A loose quick release mech will likely result in you flying off the bike. Also, test the brakes off the bike and make sure they feel strong. Then start slow, and test the brakes before picking up much speed. As long as these things are checked and working, the bike should be safe to test-ride, even if it needs work in other areas. I say should, because there are other things that are harder to spot unless you really go over the bike. For example, I’ve had handlebars and seats/seat-posts come loose while riding. A handlebar coming loose is not fun, especially at speed.
Come Prepared: Bring a multi-tool and a small adjustable wrench. This will enable you to do some important checks on the bike.
Getting Started: Something I like to do when first checking a bike is to pick up the front-end and bounce it on the front tire a bit, then do the whole bike on both tires. There shouldn’t be any rattling, except for maybe cables slapping the frame a bit.
Remove the seat-post. If it doesn’t move, you might want to walk away from the bike unless you want to take your chances at post removal. Even then, the bike should be seriously discounted because that post might never come out.
Test each part. Shifters, brakes, etc. Remember that each aspect of the bike that needs work will require time/money, and it all adds up. For example, a $50 bike isn’t a good value if it needs a full rebuild (usually just under/around $200) and $100 in parts. In that case, you might as well buy a $350, ready to ride bike.
Brake levers should only have a bit of play in them until the pads hit the rim. Brake pads should be level to the rim and contacting properly, e.g. slight toe-in, not hitting the tires, not running off the rim, etc. Don’t get on a bike with poorly functioning brakes unless you are confident you can stop the bike. Run through all of the gears. All gears should work and not make noise except when cross-chaining. All bearing units should spin smoothly without play. A gritty feeling means the unit is contaminated and currently has dirt acting like sandpaper to the bearings and bearing surfaces, which means an overhaul is in order. A bearing unit that is “stiff,” or hard to turn, either has old, dried up grease, or is overtightened. Be wary of overtightened bearings, as the unit may be damaged and need replacement. I find that wheels in particular are often maladjusted, which causes premature wear/weakened performance. Tires should be free of cracks or gouges and should hold air. When pumped up, they should not seriously deform when ridden. If they do, try pumping them up harder. Alignment: Parts should be aligned properly. Bars level, stem straight, saddle straight, etc. Some of these are easy fixes, but adjusting brake lever/integrated shift lever position on a road bike can be time consuming, particularly when you have hidden cables to deal with.
Rim Damage: Check for cracks in the rims, usually found around the eyelets. Also look for other signs of rim wear, such as the braking surface being worn down from years and years of braking use. Caution: If ridden, damaged rims should be inspected regularly.
Frame Damage and Wear: Check for dents, dings, cracks, and rust on the frame and parts. Mild rust isn’t a big deal, and is often easily removed. Caution: Dents and dings, to the frame especially, can be cause for concern, particularly on thinner, “racier” frames. Keep an eye on any frame damage, and track any progression to the damage. If in doubt, take the bike to a professional.
Smart Buying: Know what you are buying. Do your research. Check the specifications. Figure out if the bike suits your needs. Don’t be in a hurry to buy, but don’t be afraid to jump on something if you think it won’t last. The best deals are snagged within a few days (if not hours) unless they are advertised poorly.
Negotiation: Everything that’s wrong with the bike should be pointed out and the price should reflect all work needed. If the seller is hard-headed, don’t be afraid to walk away.