From my brief research of a few minutes ago, the Dominican Breviary uses the same arrangement as the 1960 Roman Breviary (which is the Pius X psalter). Therefore if you picked up a “how to” for the pre-Vatican II Roman Breviary, if one exists, then the Dominican Breviary will work exactly the same way, but with the Dominican calendar, and Dominican antiphons, etc.
As someone who has been praying the Divine Office for 15 years, I don’t recommend that folks start with pre-Conciliar, or Monastic, breviaries. They are too long and complicated, and one ends up just getting frustrated. I do use a monastic Breviary now, the same one as the community I’m attached to as oblate (all 150 psalms in a week), but it took years to build up a discipline that allowed me to reach that, and when really, really busy, I still revert to the Liturgy of the Hours.
Instead I would recommend you use the Liturgy of the Hours, which is what the Dominicans now use as well, but with their calendar. If you want to use Latin, there is a Dominican antiphonary out there. It has all of the Liturgy of the Hours, in Latin, noted for chant, using Dominical antiphons, hymns, etc. It is available on-line, here are the links:
Advent and Christmastide:
http://musicasacra.com/dominican/a1.pdf
Lent:
http://musicasacra.com/dominican/a2a.pdf
Eastertide:
http://musicasacra.com/dominican/a2b.pdf
Ordinary Time I:
http://musicasacra.com/dominican/a3.pdf
Ordinary Time II:
http://musicasacra.com/dominican/a4.pdf
If I were you, I’d print those out as booklets; a good print shop can do that for you using a spiral binding, and it won’t be too costly. Then you can use it against your existing Liturgy of the Hours so you have Latin and English side-by-side.
The Liturgy of the Hours is a beautiful gift to laity. It is so easy to manage, and very flexible. You can add more daytime hours if you want; you can pray psalms 4, 90 and 133 at Compline every day (from memory, in the dark as in tradition) if you want. You can add Vigils canticles to the Office of Readings if you want.
Add the above antiphonaries, and you’re in Divine Office heaven, especially if you learn to chant, then you can sing the Hours with the above tools.
