I am currently working on finger strength, so this is a topic that will most likely be updated along the way and for now will be relatively for finger strength training beginners. An important thing to note is that you must listen to your body when finger strength training. It is really easy to get a pulley injury from doing more than you can handle, so if your body is straining, back off.
It helps to have access to a hang board or systems board. Luckily, the gym I go to has both. The thing I like about the systems board is that it can give you a chance to work on holds that are not on the hang board. For example, the systems board at my gym has lots of pinches, but if you have a hang board like the one above you will see there are not any pinches.
The best way to start is with dead hangs. This is hanging all your body weight from the holds you chose (no feet). If you are a beginner, you probably want to try jugs first. If you are more advanced, you want to try whichever holds you can hold. When your hands are on the matching holds, you hang for up to 10 seconds. If you can hold up to 10 seconds easily, this is not challenging enough. If you fall somewhere before it could be okay. Ideally making it to at least a few seconds is probably best.
Then take a two minute timed break and get back on those same holds and try again. You may find your times change. The first may be six seconds the next eight seconds, and the third five seconds. That is fine. Keep a record of how long you can hold them. Try each set of holds four times with two minute rests between. Try three different holds, each with four reps and two minute breaks between. You could try to get progressively harder like a big crimp then smaller crimp. I like to try different holds for my three like a pinch for one, crimp for one, sloper for one, etc.
If you can use any hold on the boards and reach 10 seconds easily, try adding weight. Using a harness, weight and sling, start with a small weight and attach more accordingly. You could also get a weighted vest or ankle weights. The problem with this is that as you progress you will need more or less weight depending on which hold. If you can only use a few holds and you master them, you could try adding weight to them until you get to a point where you are strong enough to use other holds. For instance if I can use a big crimp easily, but the smaller ones are impossible. I might want to add weight to my big crimp ones until my strength grows enough that I can try the smaller ones.
You can also try offset hangs. This is where instead of being symmetrical with your holds one hand is higher than the other. Maybe your left hand is on the jug at the top, while your right hand is on a crimp at the bottom. Then switch to left hand on the crimp, right on the jug.
Rock and Ice has an article about hangboard use that might help too: http://www.rockandice.com/lates-news/knuckle-down-and-dead-hang
I am training for bouldering strength, so the short times work. If you want endurance, that article offers some suggestions. Like it says, however, it is important to decide which you want.
Will post more as progression or new discoveries are made. Let me know if something else works well for you.