I think some of the more telling things about our understanding of the Sabbath should be what Jesus said about the subject, and more importantly what Jesus did on the Sabbath.
Jesus healed on the Sabbath several times Matt 12:9-14 (NIV)
9 Going on from that place, he went into their synagogue, 10 and a man with a shriveled hand was there. Looking for a reason to bring charges against Jesus, they asked him, “Is it lawful to heal on the Sabbath?”
11 He said to them, “If any of you has a sheep and it falls into a pit on the Sabbath, will you not take hold of it and lift it out? 12 How much more valuable is a person than a sheep! Therefore it is lawful to do good on the Sabbath.”
13 Then he said to the man, “Stretch out your hand.” So he stretched it out and it was completely restored, just as sound as the other. 14 But the Pharisees went out and plotted how they might kill Jesus.
Earlier in this same passage Jesus' disciples had helped themselves to some grain in a field, also unlawful according to Jewish law. These things indicate a new understanding of Sabbath was underway.
Mark 2 (NIV) addresses this same event and has a different quotation here
27 Then he said to them, “The Sabbath was made for man, not man for the Sabbath. 28 So the Son of Man is Lord even of the Sabbath.”
In fact all of the gospels depict Jesus doing at least one if not more healing miracles on the Sabbath. All of these gospels record the Pharisees challenging Jesus for doing what they considered work on the Sabbath.
This study would seem to indicate that the rules for the Sabbath have changed. It is a day to minister to those around us, to benefit our communities. It is still for sure a day of rest, however "The Sabbath was made for man, not man for the Sabbath" should resonate with us, it's not a day that we should feel restricted by what we cannot do, it's a day for freedom and rest.
The same freedom from the restriction of the law that is preached in the rest of the New Testament is present here as well. There is no set regulation for what we can do on the Sabbath or how it should be spent. We are free. It should be spent resting or doing things in our community, but what rests me is not what rests you.