~ Kanji lessons - readings
The kanji lesson tests question you only on the readings. A reading is how the kanji is pronounced. Most kanji have more than one reading; when the kanji is used in different words, the kanji has to be pronounced using the specific reading which is correct for that word.
The 'reading' section is the main part you have to memorise to pass the kanji tests. The 'combination' section is also very important for passing the test. The 'meaning' section is just useful stuff information, but not part of the test.
~ Vocabulary lessons - memorising each symbol
I'm not sure what you mean. You don't have to memorise each kanji symbol, as you are given the pronunciation in hiragana; you can just memorise the word's pronunciation and English, and can ignore the kanji if you wish.
If you don't mean the kanji symbols but are talking about the hiragana symbols also... -- You need to learn hiragana (and katakana) before even starting the lessons.
~ 'bunch of symbols come together and make one symbol making a certain word'
I'm not sure what you mean, again. The kanji symbols are made up of smaller parts called 'radicals'. If you learn some of these, it will help a LOT with learning kanji.
If this is what you mean, have a look at this very useful page:
http://en.wikipedia.org/wiki/Table_of_Japanese_Kanji_radicals