Joseph was named for the LDS Prophet, Joseph Smith, and for the Prophet's brother, Samuel Smith. His parents were devout "Mormons", Malinda Riley and Elias Adams, who were close to the Latter-day Saint leader at Nauvoo, Illinois.

When just ten years of age, Joseph and his family, headed by their father, Elias, made the trek across the Great Plain to the Salt Lake Valley in Utah Territory. They were with the Packer family in a division of wagons headed by John Bair, the former Branch President in Garden Grove. Elias had served as President of the Pisgah Branch, so they were well acquainted.

This group traveled with Captain David Evans, who had lived near the Adams family at Payson, Illinois. The year was 1850.

In the spring of 1851, census taker, Edison D. Rich, listed the Elias Adams family and their homestead along the mountainside, north of "Kay's Ward", which would later be called Kaysville and included the east canyon on his property, which is still known as Adams Canyon.

This is where Joseph grew up, farming and raising sheep and cattle, with his father and brothers. The new Territory of Nevada had been created in 1861 when his brothers, John and Rufus, decided to try their hand at gold panning while ranging some of their sheep at Carson County, Jacks Valley, Utah Territory. Trips back and forth to and from Genoa would prove to be successful in both ventures. Younger brother, Joseph, then nineteen, went with them and freighted between Nevada and California two years before returning home to Utah.

Their father had been a rifleman in the War of 1812, was a hardened frontiersman and taught his sons many things they needed to learn about prospering and becoming gentlemen at the same time. Joseph would later help the family build the Adams Dam and Reservoir Irrigation System, still functioning today as a primary source of irrigation water delivery in Davis County, Utah.

They also built the Adams Mercantile in Layton, Utah, which would eventually house a bank. Elias was a brick mason and also taught that trade, plus business and finance techniques to his sons. During the 1860/70's, the family owned 400 acres at Geona and built a second home there.

Their Genoa home was twenty five miles from the Comstock Lode of silver ore. The Adams boys used clay from their farm to manufacture bricks for buildings in the Lode's town of Virginia City, the Carson City United States Mint and the Genoa Court House. Their two story brick home located there also provided 21 rooms to be rented out to freighters on the route to California. They were able to sell their dry farm hay to passing freighters and also corral up to fifty teams at a time.

On 6 February 1867, Joseph had returned to Kaysville and married Isabelle Smith, who had immigrated from Liverpool, England, with her English mother and her father, originally from Scotland.

They were called by Brigham Young as members of the first colonization effort in the Little Colorado River Valley, Arizona Territory. Joseph left Salt Lake City in the spring of 1873, finding "nothing but sand and rock and crooked cottonwoods". The group encamped at House Rock Valley Spring.

While awaiting the fate of the expedition, Joseph carved a petroglyph high in the Vermillion cliffs above the spring. A photo of it was taken and preserved in his personal papers. It was later reproduced upon his gravestone. The party was later recalled to Utah, where he remained for the rest of his life.

They settled at Layton, where 14 children were born to them:
Malinda Adams, Samuel Joseph Adams, Jr., Lillie May Adams, Laura Adams, Lawrence Adams, Olive Lavina Adams, Grace Adams, Asa Smith Adams, Katherine Pearl Adams, William Elias Adams, Charley Adms, Parley Adams, Elzada Lovina Adams, and Darlena Isabella Adams.

In February 1882, a new general store was incorporated at "Kays Creek" (current day Kaysville) with a capital stock of $30,000.00. It was named Kaysville Famers' Union. Joseph, his brothers, George W. and Elias Jr, along with brother-in-law, Richard Pilling, were four of the eight directors.