I guess I should add a comment since I'm doing GCSE Japanese this year! On the whole, kanji are remarkably literal when used to build "new" words from what I'd call "base" kanji. So for example, the kanji representation of telephone is the symbol for "electric" + the symbol for "talk".

However, as for readings. The "Kun" reading is the "on-its-own" reading and is as you surmise its meaning as a single Japanese word. The "On" readings, are as you say the combination of two or more. These spawn new words but crucially these are still Japanese words; they don't inherit Chinese pronunciation or grammar. The meanings however, stay intact between the two languages. And yes, it's a bugger to remember all these meaning and pronunciations!

Our teacher from last year (originally from Tokyo) mentioned that Chinese and Japanese tourists can do quite well without knowing a word of either's language since they can usually decipher signs, menus, etc. via the kanji alone.

If you're looking for interesting kanji, check out:

http://www.kanjisite.com/

It's a good site to simply browse :)