This phrasing does work better as two separate sentences:
You know, there are a lot of nice places. You can spend your time there.
As a single sentence, it goes a word too far:
You know, there are a lot of nice places you can spend your time.
"You can spend your time" is a relative clause. As phrased above, it is a contact clause, which simply means that the relating word is omitted. Regardless of whether omitted or supplied, it serves a role in its clause. It is the complement of "your time", as licensed by "spend".
When the clause stands as a separate sentence, the word "there" takes the object complement role. When the clause stands as a relative subordinate, that role goes to the relating word -- even when the relating word is merely implied. Contact clauses are recognizable because they contain an otherwise unfulfilled predicative grammatical role.
In your example, it is as if there are two words competing for that same role:
You know, there are a lot of nice places [where] you can spend your time there.
Either fits, but both don't.