You can use harsh chemical strippers to remove varnish from your wood floor or you can spend hours and hours sanding your floors to remove every last trace of varnish. Probably the easiest way to remove varnish from hardwood floors is to spread on denatured alcohol using a mop. You'll probably want two mops, one to apply the alcohol and a second mop to pick up the dissolved varnish. You will still need to sand your floors, but the denatured alcohol will save you hours and hours of work, cut the dust you'll have to deal with in half, and generally make the whole experience a lot easier and faster than it would be otherwise.
Remove all furniture from the room.
Vacuum and then damp mop to remove dust and most dirt.
Fill your two buckets nearly full of warm water. Add 2 cups of denatured alcohol to each bucket. One bucket will hold your spreading mop and the other bucket will hold your clean-up mop.
Open windows to get good ventilation in the room. Start in a corner furthest from the door and pour out a plate-sized puddle of pure denatured alcohol, straight from the bottle. Spread it around with your spreading mop.
Immediately take your clean-up mop and rub the area again, doing your best to pick up as much dissolve varnish as possible. Put your clean-up mop in its bucket of alcohol/water while you pour out another puddle of pure denatured alcohol and spread it around with your spreading mop. Then clean up that area with your clean-up mop.
Remove as much varnish as possible using your denatured alcohol. Sand the floor with your floor sander and your 80-grit paper. Make sure you keep the sander moving at all times to avoid cutting too deeply into any one spot on the floor.
Vacuum and use tack cloths to remove dust. Inspect the floor for areas that might still have varnish and need another pass of the 80-grit sander. Once the varnish has been removed sand your floor with the floor sander and the 120-grit paper, vacuum and wipe down with a tack cloth.
Wear a breathing mask when using the alcohol.
Wear a breathing mask when using the sander.