I live in a smaller two bedroom apartment, and the only source of heat is from a small gas furnace in the livingroom. We are having a baby in August and will be living here for the winter once they are born. Its really not very warm in other rooms in the apartment, such as the baby’s room.

What can I do to either make the apartment more energy efficient such as putting plastic on the windows? Which are single glass, and we do not own the place so we will not be replacing them.

Anything else?

Also if all else fails what is a safe cost efficient and energy efficient space heater?

Any electric heater would be the same in how efficient it is. The safest kind of eclectic heaters are the ones filled with oil.
Any cracks in the house should be filled in. Air infiltration is your biggest heat loss. So caulk any holes in the windows etc.