So this verb mood has the same function as some of the verbs we refer to as modal verbs: should, might and ought. They are used to say things like, "I ought to paint the house." That implies that painting the house is something that would be good to do (although I'm not doing (yet) doing it).
This verb tense (or rather mood) is called the optative. It really means something along the lines of "let's..." "Shall" when used in a regular affirmative, indicative sentence (not a question), is mostly used as a fancy alternate for "will" or a slightly weaker version of "must."
If this were a question, you could use "shall," but it isn't. It is better to use "let's." "We should" is accepted but that is because it can mean the same thing as "let's." [the semantics of modal verbs is a nightmare and they don't really line up well between Germanic languages, let along all languages]
Why is this lesson so strict about the 's in let's... It's never be so particular about little grammar mistakes before. Now I just skip the apostrophe and i get it wrong :-/