What does the Bible say Christians should or should not do on the Sabbath? Most of the Christians I know do whatever they like on Sunday: working at home, paid employment etc.

Is there any specific information in the New Testament that suggest what Christians should or should not do? 

Are we supposed to obey the Ten Commandments? What does that mean for Christians?

The Ten Commandments (Deuteronomy 5 NIV) tells us to:


  12 "Observe the Sabbath day by keeping it holy, as the LORD your God
  has commanded you. 13 Six days you shall labor and do all your work,
  14 but the seventh day is a Sabbath to the LORD your God. On it you
  shall not do any work, neither you, nor your son or daughter, nor your
  manservant or maidservant, nor your ox, your donkey or any of your
  animals, nor the alien within your gates, so that your manservant and
  maidservant may rest, as you do.
Mark 2:27 NIV
   Then he [Jesus] said to them, “The Sabbath was made for man, not man for the Sabbath. 


The new Testament does not specify that there is anything we , as Christians, should do/not do on the Sabbath.
The Sabbath was made for us to rest. In Genesis it says that God rested on the seventh day,   we should likewise also take a day to rest.