The first record we find of Anthony Snow in New England, a feltmaker by trade, is his request to the court, on 7 May 1638, for 3 acres of land in Plymouth. Abigail received her marriage portion from her mother on 9 January 1639/40, which consisted of a house and eight acres of land at Wellingsly or Hobs Hole. Anthony sold this property on 24 December 1640 for 16 pounds, 10 shillings. Five acres were granted to Anthony on 2 November 1640, and a five acre grant was recorded to him on 31 December 1641.

Jury service was frequently required of the settlers and Anthony served many times on the various juries from the petty juries to the grand juries. Even Abigail served on a coroner's jury in 1678 although women were not often called for public service.

Anthony is on the 1643 list of Plymouth men able to bear arms. At a town meeting on 10 February 1643/4, a resolution was passed that wolf traps should be set up according to the order of the General Court and one was to be made by Anthony Snow.

Anthony sold his house and the adjoining eight acres of enclosed upland in Plymouth to Thomas Dunshame on 25 August 1645 and this probably marks the time this family moved to Marshfield. He became a freeman and was appointed surveyor of highways at Marshfield on 5 June 1651. He served the first of many terms as deputy on 3 June 1656 and was a selectman for many years. Among his other public duties, he was often called on to appraise and lay out land. On 7 June 1665, Anthony was granted about thirty acres of land on the west side of the Namaskett River. On 1 August 1665, he was on a committee appointed to arbitrate about land. In 1666, he gave land for a burial ground which was later called Cedar Grove Cemetery.

Anthony and his wife Abigail (Warren) Snow had six children, order of birth unknown:

• Lydia, b. in Plymouth, MA, ca. 1640; m. Stephen Skeffe.
• Josiah, m. Rebecca Barker, daughter of Robert Barker.
• Abigail, b. ca. 1650; buried in Marshfield, 26 June 1682; m. in Marshfield, 12 December 1667, Michael Ford, son of William Ford.
• Sarah, b. ca. June 1651; m. Joseph Waterman, son of Robert and Elizabeth (Bourne) Waterman.
• a son, b. in Marshfield, 25 March 1655; died young.
• Alice, b. in Marshfield, MA, 18 Jan. 1657/8; m. Robert Barker, Jr., son of Robert Barker and Lucy (Williams) Barker.