Learning English by listening to songs is not always the best path. Like you said, there are often lines or words that are omitted. A lot of lyrics are written to fit the music. This causes many lines in certain songs to be grammatically poor. If you randomly listen to any song out there, you may learn the wrong things.

You'll have to ask some of your most trusted native English speakers to recommend songs for you to learn from. Many people may not like the genre, but Copacabana by Barry Manilow is an excellent song - musically and lyrically. You can learn structure (but not punctuation), tenses, genders, third person, and general story-telling.

http://www.guntheranderson.com/v/data/copacaba.htm

If you read through each verse, you'll find that all his ideas are complete and well structured to clearly tell his story.

Many of the older songs have better lyric writing as well. Many songs made popular by the likes of Nat "King" Cole and Frank Sinatra have well written lyrics.

Try to avoid learning from songs that use too many metaphors, reduced forms or omissions and slang.

-A

As Arvsworld wrote, lyrics can be metaphor-heavy. We get a lot of questions about lyric meaning on these boards, and the only person that could truly define the song's lyrics is the author.

That doesn't mean you shouldn't ask, but expect that sometimes the answer will be "non-standard idiom", and your guess is as good as ours.