Are your constitutional rights limited once you are convicted of a crime and sentenced to prison? Is there any clause in the US constitution that deals with this or is it simply based on crime one has committed?
Yes, many constitutional rights are limited upon conviction and imprisonment. Most notable among these is the right of liberty. The major constitutional clause is the "due process" clause of the fifth amendment that requires due process of law for limiting certain rights:
... nor shall any person ... be deprived of life, liberty, or property, without due process of law....
The fourteenth amendment extends this prohibition to the states, along with some others (emphasis added):
All persons born or naturalized in the United States and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
It would be simpler to list the rights they lose. The most obvious rights lost by people convicted of a felony are the rights to vote, the right to bear arms, and to a degree, the right to be free of unreasonable search and seizure. On occasion, a convicted felon can petition to have some of those rights restored, but said restoration is not automatic.
If you don't like that... don't commit a felony crime.
If one joins the military or gets a security clearance, they agree to give up their right to free speech, at least on subjects related to their military service or security clearance. Arguably, one joining the military gives up their right of liberty and pursuit of happiness... unless someone derives happiness from pushups and swabbing out latrines.
The accepted answer covered the 5th amendment pretty well, so I'll focus on the 1st and 8th.
First amendment rights, as far as the practice of religion, are somewhat well protected. Prisoners are allowed to practice the religion of their choice, religious texts are often provided and diet restrictions mostly respected. Free speech on the other hand is often limited by institutional rules...
The standard for restricting speech in prison is pretty broad, as established in Thornburgh v. Abbott in the US Supreme Court:
The First Amendment does not prohibit the Federal Bureau of Prisons from rejecting certain publications if said publications are detrimental to penological interests.
This combined with other case law more or less says that speech, correspondence, and reading materials may be restricted if it can be deemed potentially dangerous.
8th amendment rights are a hotly contested issue and are pretty constantly being challenged in the courts...
Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted.
Obviously this is being argued in nearly every bail hearing where a person's ability to pay is in question.
Perhaps more controversial, the meaning of "cruel and unusual" is constantly being challenged and refined. Look into recent debates about solitary confinement and the death penalty.
Cruel and unusual punishment is a charged topic and probably too broad to get into here, but most cases involving the treatment of prisoners are tied to how we define these words. Everything from healthcare, to diet, and even yard time have been informed by how courts have interpreted them.