Does a cop have to read me my rights after I'm arrested?

Short answer - No.

In TV and Movies the scene is very familiar...person gets arrested, cuffed and the police officer immediately, in a very dramatic and ominous tone begins...."You have the right to remain silent (dramatic pause)...anything you say...etc...etc."  But is that how it really works in real life?  Not really.  Sure, there may be some officers that immediately pull a Law & Order and whip out the Miranda Rights immediately after cuffing the suspect.  But many times, it doesn't happen until much later, and frankly, it's not even required to happen.  
"Reading your rights" comes from the landmark Supreme Court case of Miranda v. Arizona.  Miranda warnings are warnings meant to provide suspects with notification of important rights and protections they have before police questioning.  A person needs to be in custody and about to be questioned before Miranda warnings come in to play.  Theoretically, someone can be arrested, taken to jail, bond out and never have been read their Miranda warnings.  As long as they weren't questioned about the crime, then its unlikely the lack of Miranda warnings will be a legal issue.  Of course, there are many gray areas associated with Miranda warnings, i.e. when they were read, when the person was questioned, what they were questioned about.  But, in general, you must be in custody and be questioned about the crime for Miranda warnings to be legally necessary.  The following article summarizes the issues surrounding these warnings well.
www.findlaw.com/criminal/criminal-rights/miranda-rights-and-the-fifth-amendment.html