The website of the National Museum of Qatar seems to suggest that when it's re-opened, it'll be an amazing place to find out about the history and culture of Qatar. It should also have an impressive building! Based on what I saw of the building site when my taxi went past earlier, it's shaping up nicely, but certainly not finished...
In the mean time, is there anywhere else I can go in Doha to learn more about the history and culture of Qatar? Any other museums or cultural centres or similar that'll cover at least some of it?