How should I install a chimney for a wood stove in my garage?
Assuming open flame heaters are allowed in a garage in your area, carefully check all clearance requirements for the stove and flue.  I would encourage you to adapt to triple wall insulated flue pipe at the ceiling of the first floor and stay with insulated pipe the rest of the way. Do not attempt to change back to single wall pipe. 

You may want to install a couple of floor registers to help take some of the warm air from the first floor to the second floor. Remember, heat rises, and depending on how long you run the stove, the heat will eventually fill the second floor.  Some simple electric fans can also be placed strategically to move the warm air around and even out the temp in the building.