When lubricating your bike, really all you have to look for is the moving parts, where metal pieces move against one another. Use a light, specially-formulated bike lubricant like Boeshield T9 Waterproof Lubrication, or Pedro's Chainj, and not any old junk that you find in your garage. Oil that is too thin will dissipate quickly and not hold; oil that is too think will gum up and attract lots of dirt.
In particular, focus on these spots:
The chain: Your chain is your bike's primary moving part and the one that needs the most love and frequent lubrication. If you ride in dusty or muddy conditions, you should clean your chain regularly.
Derailleur assemblies: Your front derailleur and rear derailleur are what moves the chain between gears when you shift. These assemblies are made up of a number of small moving parts, including two small pulley wheels. You want to keep these clean and lubed so they don't bind up or become rigid. Shift the gears while you turn the pedals so you can see how the derailleurs operate, and then apply lubricant to any moving parts, including the pivot points of the assemblies.
Brake and derailleur cables: These cables control the operation of your brakes and allow you to shift gears. If they get rusty or seize up from lack of lubrication, you will not be able to stop properly or change gears smoothly, if at all. And that's a major bummer. Check them frequently, especially if you ride in dusty or wet conditions, and re-lubricate as needed with a few drops of oil.
Brake and shifter levers: Located on your handlebars, these levers are crucial for braking and changing gears. Apply a drop or two of oil to the moving points of the levers and the barrel adjusters to keep them functioning properly. Then wipe away any excess oil to keep from attracting dust.
Brake assemblies: on the brake assemblies (mounted on your frame at your front and back wheel) put a few drops of oil on any moving parts that you see. If you have trouble recognizing these pivot points, you can squeeze the brake levers, watching closely and see where they move. Anywhere these metal parts move against each other is a good place to lubricate. Be very careful not to get any oil on your brake pads. That will make it more difficult to stop quickly.
Pedals: Put a few drops of oil on the part where your pedal meets the crankarm. Again, focus on getting the oil on the moving part that rotates around the spindle, which screws into the crankarm.
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