For earring hooks, I generally use 20ga HH (half hard). That seems to be an all around good size for everyone's ear holes. 18ga seems a little bit of an overkill and can be too thick for piercings, and 22ga just seems to flimsy, even if you hammer it. I have used it in a pinch, especially with more delicate earrings, but it isn't my preference. Some wire resources offer 21ga which is also a good choice, but in all my years of wire working I've never purchased 21ga wire.
For beaded links, where you link beads together to make a bracelet or necklace, I generally use 22ga (HH) wire. If my beads are between, say 4mm and 8mm I think this is a good choice. Sometimes I like the look of a b it heavier wire with 8mm beads and might go up to 20ga (HH) if the bead hole allows it. The highest I have gone for linking beads is 18ga (DS) dead soft, and usually use that size for chunky, larger beads where I want a substantial silver look to it, otherwise I stick with 20ga (HH). So all in all, for bead linking I use 22ga and 20ga (HH) the most.
For fishhook/shepherd's hook clasps I pretty much use either the 18 or 16ga because I hammer them to harden them to keep their shape. I will use 20ga half hard on more delicate pieces and hammer them well to harden them, and that seems to work very well too, but I would not go any smaller than that for a clasp. You might be better off using a commercial spring ring clasp (which I will use myself).
For wrapping briolettes I try to use the largest wire that will go through the holes. Generally, gemstone holes are poorly drilled, 26 and 24ga wire are usually the best wires of choice. It's thin and can become brittle and break if overworked, so I usually recommend using DS. These are also great knitting and coiling wires, so I always have a few oz of each in my inventory. Pearls also have notoriously small holes, but I have been fortunate to be able to get 24ga wire through them without having to enlarge the hole. This is much easier to do than to try to enlarge gemstone holes and a topic I will cover later.
For headpins, again my rule of thumb is pretty much whatever will fit through the hole. Headpins should always be HH. Length is also a consideration, I don't like using short headpins, they should be at least 2" in length and longer. If you want to fit more than one bead on it you need to leave enough room to be able to make a loop and have enough 'tail' to wrap around a couple of times to make a secure wrapped loop. You can make a simple loop as shown in the beginning of this link: http://www.howtodothings.com/video/how-to-make-a-wrapped-loop-bead-tutorial but I personally never, ever use that type of open link. It's way too easy for the loop to pull apart and have your jewelry break. You can probably get away with it for earrings, since there really isn't any tension on it, but I'd never use it to link beads together in a necklace or bracelet.
Well, that's all I can think of for now about the basic uses of wire. I will end here by telling you what I normally stock in my inventory so I have what I need at the ready. I will highlight in red the wire that I think you, as a beginner, should start out with and you can add more as you expand in your wirework.
26ga round DS
24ga round DS & HH
22ga round DS & HH
22ga square DS
16ga round DS or HH (some places only have the DS but because of the thickness of the wire, DS is pretty strong and easier to work with and you can hammer it in some cases to harden it further).