Burundi is like it’s neighbor, Rwanda, a really beautiful country with amazing hills. Obtaining a visa for Burundi is very simple, at least in Rwanda where I applied. It took 20 minutes, 2 passport photos and 40 dollars for a 3 day transit visa. It is highly recommended to take the bus from Kigali to Bujumbura as this is one of the most scenic experiences in Africa and on great roads. Furthermore the buses are quite comfortable and really cheap. Visa on the border has been reported to be a possibility, but as only a few have done that it’s recommended to get it in Kigali.
Other embassies also issue the Burundi visa easily, but the process has been reported to take much longer and in some cases a few weeks. The embassy in Kigali is really friendly, super fast and its definitely a highly recommended option.