Once a week, I'll blog about some things that I've learned while cycling to help you make your cycling experience more enjoyable and to make you more familiar with your own bicycle. While you probably won't become a top notch mechanic by reading these, you should be able to address minor mechanical issues.
In order to address a variety of mechanical problems on the road, you only need a limited number of items, which I suggest you bring with you on every ride. Those items include: a spare tube, self-adhesive patches (unless you also want to drag glue along), a pump or CO2 inflater with 2 cartridges, a few dollar bills, a couple zip ties, tire levers and a multi-tool with a chain tool built in. With those items in hand, you should be able to deal with most unexpected bike emergencies on the road. And here's how in no particular order.
A flat tire: This is by far the most common one, but I am shocked at the number of cyclists I've encountered who have admitted to not knowing how to change a tire. My guess is that those cyclists carry a lot of cash and often take cabs when their tires go flat. The tools you'll need to resolve this issue are: spare tire (or patches if this is your second flat of the day), tire levers and pump/CO2 inflater. I can write a nice long step-by-step description to how to, but for this type of repair, a picture (video) is worth a thousand words.
This is probably the best video out there that teaches you how to change a tire. I must say, however, that changing a tire is as much of a know-how as it is an acquired skill. I highly recommend practicing this over and over again as you're sitting in front of the television watching your favorite program. It will save you lots of time on the road and your biking friends will love you because they won't have to stop for 30 minutes and watch you labor over your tire.
One note on CO2 cartridges. If you're riding on a 700x23 tire (the most common in road biking), a 16g cartridge will pump your tire to 100psi, so don't be afraid to let the entire thing loose into your tire, it will not explode. However, as the video explains, make sure you check for pinches before inflating.
A torn sidewall: Sometimes you'll hit something so bad on the road that it will not only cause a flat, but will actually tear a hole through your sidewall. If you simply replace the tube, the tire will herniate as you reinflate it and you will end up with another flat in a matter of minutes if not seconds. This is where the dollar bills come in. First, follow the steps from the video that show how to take the tire off and pull out the damaged inner tube. You will then want to proceed as if you are changing a tire, however, there is one extra step you have to take before putting the tire on all the way and inflating.
Take a dollar bill, any denomination will do, and fold it over itself twice, so you'll have 4 layers. Then, carefully insert that folded bill between the tire, where the sidewall was torn, and the inner tube. Make sure it stays in place before inflating. Now, when you inflate, the bill will hold the inner tube in its place and will prevent it from herniating and flatting. I wouldn't suggest sprinting all out on a tire that has just been repaired as such, but it should get you home safely.
A broken seat post: I guess this is a jump from the minor to the radical. It is not often that your seat post breaks. For some of you, this will never happen, but if it does, you'll be prepared. I've personally never broken a seat post, but I have broken a seat, and while it was rideable, it could very well have been otherwise. If your seat or seat post should break, take one of your water bottles, empty it and put it over your seat post. You'll have a small seat, but it will be a seat. Hopefully, you won't have too many miles to go.
Squeaky pedals: On the one hand, this isn't really a problem you need to address on the road. On the other hand, if you're on mile 5 of 200 and your pedals are driving you insane, you might want to do something about them. If you don't happen to have grease on hand, hopefully you'll have a Chapstick. Rub some of that generously on the axles where the pedals rotate, and it should keep the noise down considerably.
A damaged seat post clamp: This is a bad one. You either get it fixed, or ride standing up the remainder of the ride. If it's the clamp that broke, you're SOL, however, if it's only the bolt, there is hope. You can cannibalize the bolt that holds the top cap of your headset in place. Remember, your fork is actually held in place by your stem, so taking the cap off won't send it flying out. You will want to promptly replace the cap and bolt at home because their absence will begin to loosen your headset and that could cause damage to the bike, not to mention mess up your handling and result in a boo-boo.
A snapped spoke: This one is kind of a crap shoot. Depending on your tires and bike, it may be the end of your ride, or you might be able to gently pedal home. The first thing that will happen when you snap a spoke is that your tire will go drastically out of true and will probably rub against one of your brake pads. If your bike is like mine, with very narrow chainstays (the part of the bike that extends from the bottom bracket to the rear dropouts [place you insert rear wheel]), the wheel will also rub on the chainstay. Unlike the brakes, however, you can't really loosen your chainstays. I'm talking about the rear wheel, of course.
Assuming there is no chainstay rub, or it's the front wheel that has the snapped spoke, you should be able to make it home if you're careful. Use on of your zip ties to tie the spoke to one of it's neighbors, release your brake pads and gently pedal home. If you have any steep descents en route, I suggest going very, very slowly, unless you really want to test out the strength of your helmet. If in addition to everything else, you happen to carry a spoke wrench with you, you might reduce the effect of the snapped spoke by slightly loosening the two adjacent opposite spokes. I'm not going to go into wheel truing here because that's a blog post of its own and I assume if you're carrying a spoke wrench, you know how to use it.
A broken chain: If you have a broken chain, you better have a chain tool as part of your multi-tool, otherwise, you're SOL. But because you've read this and you know that you're supposed to have one, that won't be you. Step 1 - pick up the broken chain and examine the two loose ends. On the one side will be the male part, and on the other the female. Step 2 - you will want to remove both the male and the female parts on each loose end and as a result, the ends will reverse. Meaning the one that was male will become female (because that's the next piece in the chain set-up) and the one what was female will become male. First you will want to remove the female piece. Insert the link into the chain tool and tighten the tool until the pin is completely pushed out of the chain, leaving you with an intact male link on one end. Step 3 - remove the male piece. I'm making this a separate step because you will want to make sure to push the link out enough to remove the male link, but not completely out of the chain. You want to be left with a female end with a link pin sticking out of one of the walls. Step 4 - put the chain back on the bike. Make sure you get the chain through both derailleurs and bring the ends together. You might want to take it off the chainrings to give yourself a bit more slack, of if you're not alone, have a friend stretch the rear derailleur to make more chain slack. Step 5 - reconnect the chain. Insert the male piece into the female and reversing the process with the chain tool, you want to push the pin back, connecting the chain once again. After doing this, it is generally a good idea to flex the link at the joint and laterally to make sure it's not sticking. You should be well on your way, but remember that to save your rear derailleur, you want to stay out of the "big chainring/large cog" combination and promptly replace the chain before heading out again.
If there is an on-the-road mishap I missed and you would like me to address, leave a comment and I'll see what I can do. Obviously, way more things can happen on the road than I described, but the ones above are the ones I think are fixable. For most everything else, you better have a cell phone or a bike shop within walking distance. But regardless of the bike shop, you should always have your cell phone with you.