How to stop cold air from coming into the house
For the bottom, you need to install a (or replace the existing) door bottom (Yeah, someone was really inventive with that name).  The thing goes on the bottom of the door and will fill the gap there.

There are 2 general types, ones with vinyl fins (like this) or ones with short brushes (like this).  I personally find the latter work better, though if you have a large gap, the brush type may not be long enough to fill the gap fully, though a gap that large likely means your threshold is improperly sized.