Let's wake up early and start going on runs in the mornings!
You can say in the mornings, in the afternoons, and in the evenings. You should not say in the nights, instead you just say "At night."
Let's is really a contraction of the let and us.
When used together it typically is the same as saying "we should." Example: "Let's leave, this party sucks." or "Let's find a new solution."
"Let's go" is used in different contexts:
If someone is slow, a person could say "Let' go buddy, I don't have all day.
If you are watching a sporting event, you typically say "Let's go (team of choice)."
In older more formal dialects of English, people sometimes say things like "Let us live long and prosper." In these situations let us is used to express hope. It would sound awkward if you were to use the contraction of let's.