John Adams' cousin Samuel is seen strongly objecting to John's involvement in the Boston Massacre trial. Samuel never publicly objected to John's involvement and David McCullough speculates that he may have even viewed it as good politics.
When he returns home after the Boston Massacre, John tells Abigail that "British soldiers" fired on civilians. American colonists at that time prior to the Revolution were subjects of the king and considered themselves to be British. Also, there were only one kind of soldiers in Boston at that time. For those reasons it would have been redundant for a British subject to refer to them as "British soldiers" and not just soldiers.
At the end of Episode 1, when John Adams is leaving to attend the Continental Congress in Philadelphia in 1775, Abigail is very pregnant, presumably with their son, Thomas. But Thomas was actually born three years prior, in 1772, and it's too early for what would be the (still) birth of Elizabeth, which was 1777. The Adamses had four living children in 1775, not three.
The hymn tune that is sung, "Chester," was not written until 1778.
When telling Abigail about the events of March 5, 1770, John Adams says, "British soldiers fired into a crowd on State House Way." The Boston Massacre took place outside of what is known at the time as the Town House (known today as the Old State House), and was located on King Street (now known as State Street). State House Way does not exist.
During the Tar and Feather scene, Hancock wore a gray wig, but he wore a white wig or auburn wig as his natural hair color.