First let me say that the closest I have come to the actual city of Doha is the airport. I transited there is September last year en route to Europe whilst flying with Qatar Airlines. Coming down the ramp from the plane the city sits a tantalising distance away on the horizon.
If like me you did not have time to make Doha a stop, what you will see is Hamad International Airport.
It is huge, very modern, and very expensive. A small Greek salad and a small orange juice cost me over USD20 so be warned. If you do have time and you are lucky, you can join a free city tour. By lucky I mean you cannot pre-book, they only run at set hours and if they are full you can’t go. If the hours don’t work because of your flight times you are going to get a comprehensive look at the airport facilities.
The trains to the far concourses are fun and will give you something to do after duty free looking gets boring and they do have nap rooms to stretch out in.
If you have a long layover you can book a tour of the city. The company will pick you up from the airport and you will have a chance see the city. The free city tour, the museum of Islamic art and the souks will give you a brief introduction to the city. If you actually make Doha a stop you can obviously see much more.
As with all the Emirati states a trip into the desert will connect you with what is still the heart and soul of the country. The silence, the vastness of the desert forces you to re-examine your relationship with the natural world. It is salutary and humbling.