The best way to pick up mechanic skills is by reading a good repair book. We offer these in our store and you'll find them instructive and easy to follow. It can be scary to get started so we recommend learning the basics first, which include such things as airing tires, lubing the chain and fixing a flat.
If you get into repairing your bike gradually like this, you'll develop your confidence one step at a time and build your tool kit purchasing only those things you need right now. As your requirements change and you want to tackle more challenging procedures, you'll have an idea where to start from you previous experiences, and things should go smoothly.
Keep in mind that you only need to be an expert on your bike(s), so there's no pressure to be an all-knowing mechanic. Most bicycle parts are quite user friendly and are easy to diagnose and repair with inexpensive tools, some of which you may already have, such as pliers, screwdrivers and adjustable wrenches. Another cool thing is that a lot of bicycle parts come in pairs, so as you work on one component, you can refer to the other to see if you're forgetting anything.
To give you a head start, here are five important bike-repair rules:
1. Almost every part on a bicycle is regular thread, meaning you turn to the right to tighten and to the left to loosen (remember "righty, tighty; lefty, loosy," if it helps). The one part that's not regular thread is the left pedal. Turn it counter-clockwise to tighten, clockwise to loosen.
2. Because lightweight materials are used, bicycle component threading can be delicate so it takes a while to learn how tight is right. It's best to tighten things until they're snug, test to see if the part is tight, retighten if it needs it, and so on. If you're too heavy handed with your wrenches, you'll strip parts quickly.
3. Never force anything. If it doesn't fit, determine why before you ruin the part or damage your bike.
4. Allow time to complete a procedure. A rushed job is usually a botched job.
5. Work where you can safely leave things if you get interrupted, so important parts don't get lost. And, use good lighting and work on a smooth, clean floor so you can easily find small parts you drop.