Instead of throwing out our egg cartons we save them for crafts and fire starters. The cardboard egg cartons burn well.
Cotton makeup remover pads a fast and easy DIY fire starter. Just dab on a bit of Vaseline/petroleum jelly and light it up.
When we do get newspapers, we set them aside. Then the youngest kids can gather small twigs and wrap them in the paper for a cheap and easy DIY fire starter.
Dip them in rubbing alcohol or even warm wax to make them burn faster and/or longer.