In particular, Augustus seems to have wanted to encourage people to get married. He instituted a tax on unmarried adult citizens, regardless of gender, something that was not terribly well received. He also loosened the restrictions on who could marry whom, restricting only senators, their children, and grandchildren from marrying freedpersons.*
Related to these were his laws relating to childbearing. If a freeborn woman had three children, she would be considered sui iuris and would no longer require a guardian. The same applied to freedwomen, though they had to produce four children rather than three.
Lastly, Augustus was very concerned about the problem of adultery. His laws go into detail regarding who can kill someone they catch in the act and under what circumstances they may do so. Anyone convicted of adultery was to be punished with a loss of goods and exile to an island.** A husband whose wife was convicted was expected to divorce her or be prosecuted as a pimp.
[Julia Augusti Filia, 1553, Guillaume Rouille, Source: Wikimedia Commons]
*Given the Roman distaste for the idea of a woman marrying someone lower in rank than herself one would assume this applied only to men marrying freedwomen. However, during the time of the Empire there were a number of free women in de facto marriages with slave men who then married them on manumission. This law may have applied to them as well. For more information on such unions, see this post.
**The jurist Paul even makes sure to note that two people caught in adultery with each other should be sent to two different islands.
Primary Sources on the Julian Marriage Laws - Women's Life in Greece and Rome (Diotima)
Augustus' Laws; Lex Iulia and Papia Poppaea (18-9 BC) - University of Houston
Lex Iulia - Wikipedia