Using a smaller low budget cooker like the CharGriller you use as an example, I would steer clear of logs and use fist sized chunks. The firebox and cook chamber just arent that big.
The big key to avoiding that big burst of white smoke is to gradually introduce dry well seasoned wood to an existing HOT fire. This is the part that takes a bit of skill and artisty but it no that big of deal if you practice it a few times.
As far as your fuel, if you aren't cutting and spliting yourself, or if you dont have any experience doing so, check out Western BBQ Products
They have a wide range of products from chips to chunks to logs in just about any flavor or species you might want.