If, for example, I wanted to say "I like the book that my sister gave me", would it be
I'm using Genki to study, but they don't seem to have any examples of this particular structure that ends in "が好きです".
Yes, 「姉がくれた本が好きです。」 is correct.
The first が, because it is in a clause that modifies 本, can be also be swapped with の. So 「姉のくれた本が好きです。」 is also correct, and has the same meaning.
This is not the only reason that が can appear multiple times in a sentence.
Predicates like 好きだ are called "double-ga" or "affective" predicates1. This class also includes 分かる, いります, and できる, and all of these take が instead of を for the equivalent of the English direct object. So if you ask,
The answer could be,
This has three が, and only one of them is in a relative clause. This also implies that the other people don't like the book.
1: This is the terminology used in "Japanese: The Spoken Language". Your text may use different terminology.
Yes. You have two nominative noun phrases in your English sentence, one in the main clause and one in the relative clause. Japanese is no different.