Getting around Doha is not really hard since there are a couple of services that run around the city.
There are two kinds of buses in Doha: public bus and shuttle bus. The public bus is operated by Mowasalat and run a large network of routes which go around the whole city of Doha and some of the surrounding towns. You can pay the fare in the bus (make sure that you have the exact amount) or you can get a smartcard which you can by at the bus station. You can see the timetable online and the routes at the bus station. The shuttle bus is also operated by Mowasalat and has two routes in the West Bay area. You can download the route online. This service is free.
By tour bus
There is a company called Doha bus which offers a hop-on, hop off bus service with a number of stops between the Marriott Hotel and the Pearl Qatar. There are buses arriving at the stops every 20 minutes. Tickets are valid for 24 hours and there is a different fare for adults and for children. You need to book a ticket if you are planning on using this service.
There are two taxi services in Doha, both operated by Mowasalat: Karwa and Al Million. There are also limousine taxis, but they are not marked, much more expensive and might not use a meter. The demand for taxis in Doha is great so you may have to wait a little while to get a taxi. It may be best if you booked a taxi up from, but the taxi that you booked might not appear as well. You can find a taxi for sure at the major malls, international hotels, and the airport. Make sure that the driver uses his/her meter; if not, agree on a price before setting off.
There are several rent-a-car agencies in Doha at the airport. They are kind of hard to find without many signs. It would be best if you check out the agencies before going to Doha and maybe even reserve in advance if you are planning on rentng a car for sure. Make sure that you have an international drivers license and check the laws before going to Doha since they change fast. Driving is on the right side with similar traffic rules as anywhere.