I'd like to know the difference between the usage of coming up and coming. For example, the Pink's song says: "I'm coming up so you better get this party started".
If someone removes the "up", would it change the meaning?
The words "coming up" are a commonly used set phrase, if not quite an idiom. In the sense used in the question, they mostly indicate that someone is coming to a higher floor, particularly to an apartment from street level.
Buzz the door open because I'm coming up.
On the other hand, "I'm coming" in a comparable sense means merely that the person is on the way, and does not say anything about floors.
I'm coming -- I'm only five blocks away.
As mentioned in a comment by J.R. "I'm coming up to" can mean "I am traveling to" when the direction of travel is at least roughly north, and so "up" on a standard map. In this case "I'm coming to" can also be used with the same meaning, but the form with "up" is very commonly used by native speakers.
- I'm coming up to Toronto next month.
- I'm coming to Toronto next month.
Both of the above sentences are correct, and mean the same.
The word "coming" can also be used in several other senses, not all of which would have a parallel or related form using "coming up"
"I'm coming up" could also be used when the destination is on a hill, but that would be significantly less common.
The phrase "coming up" can also be sued to mean "happening soon, as in
The Fourth of July is coming up.
In this sense "coming" could also be used, but "coming up" suggests closer proximity in time, and greater urgency.
"Coming up" is also used of a diver or a submarine moving toward the surface of the water. Here there is no parallel use of "coming" without 'to".
"Coming up to" can be used to mean "approaching" or "getting near to", as in:
I'm coming up to the narrow bridge on Route 5.
Here "coming to" can also be used with pretty much the same meaning.
"Coming up" can also mean "getting mentioned" as in:
The issue kept coming up in our conversation.
Here there is no similar or parallel use of "coming" without "up".