I'm sure your birds would really appreciate a new larger cage. Finches need cages that are longer than they are tall so they can fly from one end to the other.
Also, instead of buying sandpaper sheets for the bottom of the cage, you could just cut newspaper to the proper size and use that. There isn't any reason for the sandpaper sheets any more than the sandpaper perch covers.
For food, you can start giving them tiny pieces of cut up fruits and vegetables as well as hard boiled egg, potatoes, or almost whatever you might be eating. They can eat much like budgies, except that the pieces should be smaller.
Once you have a cage with enough room for a nest, you can purchase one and put it up. For nesting material, they will use shredded paper, hay, dried grasses, etc. Of course once they have babies, you'll have to start thinking about what to do with the babies.