Which part of the Torah portion is chanted on Shabbat each week?

I am learning Torah cantillation (trope) by learning each week’s Torah portion. Is there a suggested part of the parashah to chant each week? I see that the parashah is broken down into seven aliyot, but I’ve never heard my rabbi chant that many verses.

It is customary to chant the entire parashahParashahפָּרָשָׁהTorah portion. The five books of the Torah are divided into 54 parashiyot or portions. Each week, Jewish communities read one parashah (singular of parashiyot); in this way, Jewish communities read the entire Torah over the course of a year.  Depending on the calendar, some weeks will feature a “double-portion.” The name of each portion is taken from the first few significant words of the portion; plural: parashiyot  (divided into seven aliyotaliyahעֲלִיָּה"Going up." The honor of being called to recite the blessings before and after the Torah reading. Also refers to immigration to Israel, to "make aliyah" to Israel; plural: aliyot. Lit. "Ascent." ) on Shabbat. Some congregations follow a triennial cycle, reading one third of the weekly parashah until, over the course of three years, they have read the entire Torah.

In Reform congregations, it’s common to read a selection of verses from the parashah or weekly Torah portion rather than to chant the entire parashah. Often the spiritual leader selects the verses, and different verses from the parashah may be selected each year. Sometimes rabbis or cantors choose verses they plan to teach or preach about, or verses that are meaningful to the children becoming b’nei mitzvah. The Reform Movement does not suggest communities read particular aliyot or particular verses over others. 

Perhaps you’ve already asked her, but it would be interesting to learn how your rabbi chooses the verses that are shared each Shabbat.