Some examples:
"Real quick" we often say to emphasize that the action will not take much of the listener's time. It is slang, but very common.
"Can I ask you a question real quick?"
"I've got to run to the store real quick. "
We would NEVER use it to say something is moving fast.
"Really quick"
Also slang, but it can mean that something is moving fast.
"She really quick getting ready this morning. "
"I've got to run to the store, but I'll be really quick." (See the slight difference in use? )
"Real quickly" is a bit uncommon, you'd more likely hear "Really quickly", which is slightly more formal.
"Okay, you can use the bathroom, but do it real(ly) quickly"
"She brought us our food real(ly) quickly."