If you look up "article usage" in a detailed English grammar book, you'll probably find a few pages of specific examples. Trying to figure out which (if any) article to use is like trying to figure out what "get" means - basically, it really just depends on context. The best way to learn them well is to just read/listen/watch a lot of English stuff and pay attention, and practice, as I'm sure you know the basic rules. Here are some examples I used to use when I taught English, though: "Open a window" = there is more than one window in the room, open any one. "Open the window" = there is only one window OR there are multiple windows but it's obvious which one I'm referring to. "The planet has a large moon" vs. "The moon is bright tonight." Our moon is "the moon" and other planets just have "moons." Same goes for "the sun" vs. "suns" in other solar systems. An individual lake or mountain has no article: Lake Erie, Lake Huron, Lake Superior, Lake Michigan, Lake Ontario. However, the group takes "the" - the above lakes are all a part of The Great Lakes in North America. See also "Mount Everest" but "The Himalayas." As you can see, it gets really specific. It's very difficult for a native speaker to explain all the uses because we just know what "sounds" right and this is one thing that foreigners very often mess up but natives virtually never do (as opposed to grammar and spelling, which we are also terrible at). So keep a grammar reference book around but be patient and try to feel which is right.

It's a very good question! I started to learn Russian and lo! What did I see? No articles! And you are right. They are just about as useful as hen's teeth. But, many languages have them. I think you can compare them, if your algebra is good, to the following: 1,2,3,4,... = the (definite) a, b, c,.. x, y, z = an, a (indefinite). Well, do we really 'know' 1,2,3,4,... we suppose so. They are pretty definite. And do we really 'know' a, b, c, ..x, y, z.... we think not. They are more cloudy. Anyway, learn them in English and you will have a good nose for them in German, French, Spanish, Italian...