Learn English – about the song “Another Day of Sun”

This is a standard idiom in English. The earth here is a metaphor that symbolizes the impossible, the immoderate, or the unfulfillable:

promise (someone) the earth (or moon)^NOAD
make extravagant promises to someone that are unlikely to be fulfilled:
interactive technology titillates, promises the earth, but delivers nothing

The origin of this idiom is probably from the third temptation of Jesus (in the Bible), where the devil tempts him with the ridiculous promise of all the kingdoms of the world:

Again, the devil taketh him up into an exceeding high mountain, and sheweth him all the kingdoms of the world, and the glory of them; And saith unto him, All these things will I give thee, if thou wilt fall down and worship me.
_{— Matthew 4:8–9 (The Bible, King James Version)}

"What's in store?" means "What's going to happen in the future?" rather than "What is being kept back?" Sometimes there is an implication that the things "in store" are planned by someone, but not always.

Edit: I forgot about "Stick to my tune." That is a piece of poetic writing which takes advantage of being part of a song. "Stick to" here means "keep to" or "stay with" in a metaphorical sense; the lyricist is saying that he should carry on doing what he normally does (his "tune" in this metaphor). It's more commonly found in phrases like "Stick to your guns".