The last time I visited Qatar was 2006 and a lot has changed. The population has increased, which means much more traffic for such a tiny country but the addition of museums and touristic attractions is amazing. You can start off by taking a look at their touristic website to see what there is to do. I didn't have enough time to do everything so a second trip is definitely in the making. The airport is simply amazing, modern and a great way to start off in Qatar.
The Qatar skyline
The country is surrounded by water which makes for an amazing corniche. The corniche surrounding the Islamic Museum is beautiful. It is green area with trees and grass and the best part is not seeing any trash. The country is kept very clean which proves it can be done if people truly care about their country. Of course, there are still undeveloped areas with the old cars and buildings and I'm sure a touristic view of the country is different than those of the people who live there.