What should I capitalize? I'm not sure!?

Both of these phrases should be capitalized. Anything that is the actual name of a place, organization, etc. is capitalized. If the name of the building was "The Capitol", and building was not an official part of the name, then you wouldn't need to capitalize building. Likewise, if the sentence read "They decided that their town needed some form of general council." then you wouldn't capitalize, because the sentence is only speaking about general council as a concept or type of group. But in your example the group actually exists and is named the General Council.