1)  We are put carbon into the air by burning fossil fuels.  
What are fossil fuels?  Plants that were buried eons ago.  
How can we reverse the trend?  By burying plant matter, thus sequestering CO2 from the air.
Fortunately, there is a convenient way to do so. Put lawn waste and paper in a landfill.
 Paper is mostly made from tree farm trees. They'll replant to grow more. New trees absorb CO2 from air.
  
   a)  Bury paper. Don't recycle it.
   b)  Bury lawn waste, don't mulch or compost.

2) Since people do burn stuff to keep warm, travel, and do stuff in general, is there a way to do this without as much effect on climate?
Yes.
Global COOLing happens when particulates (smoke) reflect some sunlight.  
So the laudable efforts to produce cleaner emissions from cars, powerplants, etc, also prevents the effect of global cooling.  
Ironic that  a filthy smoke belching 1800s smokestack which lead to respiratory problems, etc, would also put out particulates that block some sunlight.
  
  a)  make smoke, not clean exhaust.

I welcome responses whether you agree or disagree.  To be perfectly honest, I'm not sure if I agree myself ! Just putting it out there.