How does a layperson go about praying the Divine office if one is interested?
From my own personal experience, the regular recitation of the Divine Office necessitates a preliminary study of the rubrics, the structure, and the content of the Office (the Hymns, Psalms, Canticles, Lessons, Responsories, Collects, etc.), as well as the devotional and spiritual aspect of the Divine Office. Of course, I did it by trial and error method since the rubrics have been difficult for every generation of clerics, and much more shall they be for mere layfolk like us.
It is recommended that layfolk begin with either Prime and Compline of the Roman Breviary, or with the Little Office of the Blessed Virgin Mary, since these are relatively simple and unvarying in comparison with the Divine Office as a whole. Perhaps the practice of saying parts of the Office on Sunday in order to hallow the Lord's Day would be a great way to begin. It would make a wonderful preparation or thanksgiving for Holy Mass and Holy Communion, as this is how the Priests prepare themselves for such divine Mysteries.
There is also the Office of the Dead, which is a great way to begin the cultivation of the regular recitation of the Divine Office, since it is a very ancient and simple form of the original Roman Office.
Are there set times during the day?
Traditionally, the Canonical Hours were recited as follows: Matins and Lauds very early in the morning so that Lauds ended at sunrise; Prime at about 6 AM; Terce at about 9 AM; Sext at about 12 PM; None at about 3 PM; Vespers at about 6 PM; and Compline at about 9 PM. Clerics and Religious (if allowed by their Rule or Superiors) were allowed to anticipate Matins and Lauds on the previous evening in private recitation if they foresaw that the recitation thereof on the following morning would be very inconvenient or unfeasible.
However, this scheme is usually impractical for the average layman, so most layfolk who recite either the Roman Breviary or the Little Office of the Blessed Virgin, do something like the following: they may begin the morning with Prime and Terce; say Sext and None sometime around lunchtime; recite Vespers and Compline together in sometime in the early evening; and anticipate Matins and Lauds on the later evening, or recite them very early in the morning, followed by the recitation of Prime.
This is at least what I did when I recited the Hours regularly at the University (I mostly omitted Matins for paucity of time).