"okuyalım" is a suggestion, so it comes in the form of a question "shall we read?" or a more suggestive statement "Let's read". It would be nice to read together, but it's ok if we don't...
Your sentence, "we shall" would be a future tense statement, because it has a little more intention than suggestion, if that makes sense: "okuyacağız". :-)
However, with all that being said... Duo has a tendency to reject "shall" (to the frustration of many). But know that it is correct in the real world :-)
Aha is it because the mot literal translation would be: Let's read THE newspaper which is today's?
When do you use ki and possessive case?
I see how "Bugünkü gazeteyi okuyalım." is correct, but can I also say "Bugünün gazetesi okuyalım." ??
"Bugünkü gazeteyi okuyalım." Translation: Let's read today's newspaper.
Let me refine your answer which is correct:
"Bugünün gazetesini okuyalım." - Let's read today's newspaper.
The genitive & possessive suffixes. Your answer means literally: "The newspaper of today."