The Tire is Flat
After you pump up the tube, it never holds any air, slowly becomes flat again or it may be difficult to pump air into it initially.
If the tire is unable to hold a steady pressure, the tube is likely popped. Steps to replace a popped tube can be found at this replacement guide.
If no matter how hard you pump the tire won't fill up, the valve may be faulty. Since Presta valves can't be fixed, you will have to replace the whole tube. But before throwing away the tube, double check make sure the bolt on top of the valve is slightly loosened, and that the pump being used is meant for Presta valves (many pumps are meant for Schrader valves only). If you are unsure how to check either of these, you can take a look at the guide above demonstrating how to replace a tube, which demonstrates how to pump up a tire towards the end of the guide.
Drivetrain Does Not Turn Smoothly
As you pedal, you feel extra friction or hear grinding through each stroke, or the chain continually becomes stuck and binds up on the chainring.
Over time the teeth made from soft alloy metal on the chainring can wear out and become rounded, and the ring itself is commonly damaged from coming in contact with rocks. As a rule of thumb, chainrings generally last about 3 times as long as the cassette. This guide will walk you through the steps to replacing a chainring on your bike.
Rim Does Not Spin True
If the tire appears to wobble as it spins, or your tire is scraping your frame, the rim may need to be straightened.
If the tire is not set properly in the frame, it may appear that the rim is bent, but before attempting to straighten the rim, check to make sure the axle is properly set in the frame. Try removing the tire, removing any debris, and re-installing the tire.
Depending on the degree of the damage, this may warrant a new rim, but you can likely repair it by following the steps found in this guide.
Bike shifts into the wrong gears or won't shift at all.
Front Derailleur Adjustment
Shifting issues related to the forward gears, which are controlled by the right shifter, can be adjusted by following this guide.
Rear Derailleur Adjustment
Shifting issues related to the rear gears, which are controlled by the left shifter, can be adjusted by following this guide.