How to learn to chant the Liturgy of the Hours? I went to solemn vespers for the first time in my life last Friday and it was super awesome.  I brought my liturgy of the hours book with, thinking I might need it, but they provided a card with the whole thing on it.  Complete with some sort of music part where one word was underlined and it corresponded to what note to change  to.  So it was like "Lord come to my assistance" and "assist" was underlined because the pitch changed there.

So, my question is, since my breviary doesn't have that cool stuff in it, how does one learn how to do it?  For both personal recitation and for leading small groups in it?  Are the pitch changes solely based on the meter?  And why do we switch people intoning the words every other line? When doing small group Liturgy of the Hours, we usually break it up into the paragraphs.
I am glad you enjoyed sung Liturgy of the Hours.

The "music part where one word was underlined" is called the mediant, and the music changes accordingly with that word (or syllable).  If you are working with a text that does not have the mediant identified (such as a common breviary), you determine it on the basis of meter and spacing.  It is only possible to understand how to do this with accuracy once you are familiar with the tone.  Usually praying often with a community teaches you how the community finds the mediant, and you naturally learn how to do it on your own.

A "tone" or "psalm tone" is a music setting for the Liturgy of the Hours.  There are many different sets of tones and styles of chanting.  These can differ based on geographical location, language group, religious order, or other factors.  One common set of tones in the U.S. is the St. Meinrad Psalm Tones (which come from St. Meinrad's Benedictine monastery).

Different groups and psalm tones will also break up lines differently.  The basic back-and-forth, two-party structure of the chant is called "choral recitation" (because choir-style seating has traditionally been used to represent the two sides).  The simplest approach is to break up every other line, but it is more common to switch sides after each pair of lines, or especially after each stanza.  The difficulty with the last option is that different psalm stanzas have a different number of lines, so psalm tones usually have ways of handling stanzas of different lengths, ways which often include a flex line that can be added for stanzas with a odd number of lines.

More info: Reciting Tone.