Ramadhan is the month of Qur'an. We should recite as much as possible, day and night. Our pious predecessors used to recite the complete Qur'an each day and each night during Ramadhan. We may not able to do the same, but we should at least try to recite as much as possible.
Other than reciting the Qur'an, I recommend everyone to read the stories of the Prophets and their companions. The Sahabah were selected by Allah :swt: to be role models for the rest of the Ummah. You can find a very good book on their lives here:
https://islamicbookslibrary.wordpress.com/2011/07/20/hayat-us-sahaba-urdu/
It is very sad to see people of our generation knowing each and every detail of fictional characters and celebrities but they don't know much about the Sahabah :rahm:.