Unfortunately motorcyles are banned from all Botswana National Parks and Game Reserves. The police, army and game rangers have the power to confiscate any vehicle used illegally in reserves - and they do (at gunpoint).
You can ride along the transit route through Chobe, which is a wide gravel road surrounded on both sides by dense forest, running between Kasane and the Namibian Border at Ngoma Bridge. You will probably come across elies in there, but won't see that much. The best option is to stay at the Chobe Safari Lodge (camping available, but book ahead) in Kasane and take 4x4 and boat trips along the riverine area, this is also where most of the game is concentrated.
Other places worth visiting or staying at are the excellent Mokolodi Nature Reserve in Gaborone (chalets, camping, Rhino tracking), the town of Serowe (has a reserve with Rhinos), the Magkadigkadi Pans (try Nata Lodge or the nature reserve at Sua), camps along the edge of the Okavango, including the fishing camps in the North West, and possibly the Aha Hills (not sure about the status of this, but would be an interesting ride).
You could also enquire with owners of private reserves in the Tuli - there's a nice one called Mawana.