From http://www.lds.org:

What can we learn about serving others on the Sabbath from the ways the Savior served and blessed those around Him? Encourage members to review and discuss Matthew 9:10–13; Luke 19:1–9; John 11:32–46; 13:1–5, 12–17; and 3 Nephi 17:5–10. Ask members to think about these scriptures as they consider how they can serve on the Sabbath. For example, they could serve familymembers, reach out to individuals and families they minister to as home and visiting teachers, work on family history, visit the sick, or share the gospel. Perhaps members could have a family council to plan ways they can serve others on the Sabbath.

Jesus Christ taught, “It is lawful to do well on the sabbath.” As we follow His example, we can serve others—even on our day of rest.

The Sabbath provides opportunities to serve not only God  but also others.

‘Jesus reaffirmed the importance of the Sabbath day devotion, but he introduced a new spirit into this part of worship. (See  Matt. 24:20 Rather than observe the endless technicalities and prohibitions concerning what should and should not be done on the Lord’s day, he affirmed that it is lawful to do well on the Sabbath. (See  Matt. 12:12) He taught us that “the Son of man is Lord even of the sabbath day”  Matt. 12:8) and introduced the principle that “the sabbath was made for man, and not man for the sabbath”  (Mark 2:27). He performed good deeds on the Sabbath, such as healing the man with palsy (see  Mark 2:1–12) as well as the man with the paralyzed hand (see  Matt. 12:10–13) So the divine mandate of Sabbath day observance in our day is now more of a manifestation of individual devotion and commitment rather than a requirement of civil law.’ (James E Faust, General Conference, October 1991)

Mark 3:4 And he saith unto them, Is it lawful to do good on the sabbath days, or to do evil? to save life, or to kill? But they held their peace.

‘The Savior’s questions, “Is it lawful to do good on the sabbath days, or to do evil? to save life, or to kill?” set the stage for a vivid teaching moment (Mark 3:4–5). Some rabbis had determined that only in emergency situations was it permissible to save a life on the Sabbath. Of course, it would have been just as unlawful to murder on the Sabbath as on any other day. Yet ironically, after the Savior did good by healing the man’s hand, the very people who thought He was guilty of breaking the law—and who imagined themselves the keepers of the law—began on that Sabbath to plot how to destroy Jesus (see Mark 3:6).

The Savior’s words help us understand that the Sabbath is a day for doing good. One way to “do good” on the Sabbath is to “go to the house of prayer and offer up thy sacraments … , and to pay thy devotions unto the Most High” (D&C 59:9–10). Other ways of doing good on the Sabbath “may include praying, meditating, studying the scriptures and the teachings of latter-day prophets, writing letters to family members and friends, reading wholesome material, visiting the sick and distressed, and attending other Church meetings” (True to the Faith,146).’ (Institute New Testament Manual)

‘After the Savior healed the woman who had suffered a physical affliction for 18 years (see Luke 13:10–17), the leader of the synagogue where the healing occurred complained because the miracle had been performed on the Sabbath day. The Savior’s response taught that it is lawful to do good things on the Sabbath day and that those who teach otherwise are hypocrites. From this response, we learn the importance of looking for good things we can do on the Sabbath day rather than becoming overly concerned with what we cannot do.’ (Institute New Testament Manual)

 “We know from these inspired words that even the most extreme acts of service fall short of the ultimate ‘profit’ unless they are motivated by the pure love of Christ. If our service is to be most efficacious, it must be unconcerned with self and heedless of personal advantage. It must be accomplished for the love of God and the love of his children.” (Dallin H Oaks, Pure in Heart [Salt Lake City: Bookcraft, 1988], 47.)

How can we serve others on theSabbath day?