Shall we go to the cinema?
And one more question today :).
When you ask for something and make suggestions, do you use "shall" like in the sentences below?
"What shall we do today?" (asking)
"Shall we go to the cinema?" (suggestion)
Or is it more common to say these sentences in the following ways (below)?
"What will we do today?" or "What are we going to do today?"
"Let's go to the cinema!"
Richard has a great answer - just wanted to add a colloquial note.
'Shall we' is a bit formal for common usage (or it is in my experience). Among younger people on the west coast of the US (California, Washington, Oregon), we'd simply say, "Want to go to the movies?" or "Let's go watch a movie."
Similarly, "What shall we do today?" is not incorrect, it just sounds a bit funny in everyday speech. I'd say something like, "What do you want to do today?", or even, "What do you want to do?"
I'm from Virginia. Informally and among friends, we say, "Wanna go see a movie?", "Let's go to the movies!", "Let's go see a movie.", or "Wanna go see what's playing at the movies/theatre?"
Your uses of "shall" are fine.
In my English (SE Pennsylvania) "What shall we do?" and "What will we do?" might differ in the following way:
If I ask you, "What shall we do today?" the meaning is similar to "What do you think we should or might do today? What is your idea or suggestion?"
If my high school students enter the classroom and ask me, "What will we do today?" or "What are we going to do today?", the question is more like "What plan do you have for us today?" They're not asking me for my idea or a suggestion; they want to know what I have planned for them.
If I ask you "Shall we go to the movies?" (we say the movies here, rather than the cinema), I might be suggesting that my idea is that we go to the movies and/or I might be asking for your opinion about the suggestion that we go to the movies.
If I ask you "Will we go to the movies?", I am likely asking whether there is a plan in place for us to go to the movies.
If I say "Let's go to the movies", I am proposing that we go to the movies.