The Qatari government has worked hard to make Doha an educational centre in the Middle East. One of the benefits of this is the Doha Debates (http://www.dohadebates.com/), where top political and academic minds in the Arab world come together to discuss difficult issues in the Arab World. Past debates have discussed whether Palestinians risk becoming their own worst enemy, whether the Sunni-Shia conflict damages Islam's reputation as a religion of peace, or if Muslims are failing to combat extremism.
The debates are always very thought-provoking and a good window to understanding the current state of the Arab world. Tickets are extremely limited but can be obtained from the website above.
A typical Middle Eastern activity in the afternoons is to find a sheesha cafe and smoke some fruit-flavoured tobacco. One of the best places in Doha is Ras-Naswa at the non-Sheraton end of the Corniche. Located in a picturesque old-style building reminiscient, in colour and texture if not grandeur, of the red Mughal structures in India, Ras-Naswa has a nice outdoor garden and serves decent Middle Eastern food.