How to Repair a Mountain Bicycle

Assess the damage. There are three main systems of concern on a mountain bike: braking, tires/wheel and shifting. Each one of these should be looked over for maintenance and repairs.
Damage to the wheel sets can be checked by giving each spoke a short tap with a wrench. The corresponding sound should be a sharp ping and all the spokes should be in a relatively similar pitch. If this does not occur, repairs need to be made. A flat tire is an obvious repair but it is also important to check for large cracks in the sidewall of the tires, which would also warrant repair or replacing.
The braking system is very important to the function of the bike. If you notice slow breaking or abnormal noises from your brakes, it is time for them to be repaired. Make sure and note whether your bike has rim brakes or disc brakes as maintenance and repair for both is different.
If you notice slipping, slow shifting or too much squeaking in your gears, it is necessary for you to repair this area of the bike. There are two main components of the shifting system that should be checked for damage and proper function: the chain and the shifter cables.

Shop for parts. Most of the parts for simple repairs such as chain replacements and tubes can be found at your local bicycle shop, but some specialty parts can be found for less online on sites such as Ebay and Craigslist. If you are doing several repairs, this may be a good time to upgrade parts like wheel sets, cassettes, brackets and brakes.

To repair wheel sets and tires, re-tighten all the spokes that do not yield a sharp pinging sound or replace your wheel set for an upgraded one. Install new tubes on flat tires or replace tire if the sidewall has large cracks or there are several knobs missing. Fill tubes to approximately 35 psi (pounds per square inch) or until the full weight of your body can be supported without the tire spreading.
Rim brakes use one pad of rubber on either side of the rim to pinch it and cause the bike to stop. Pads must be replaced if they are excessively worn or don't have parallel contact with the rim. For disk brakes, make sure the pad is close to the disk and that the thickness is good. Replace pads when necessary and beware of contamination from lubricants.
For shifting, clean and lubricate shifter cables by removing the cables and housing from the bike. Clean and lubricate the chain and replace if broken.