Fabric wrapping
Wrap gifts in a scarf, tea towel or a square of fabric tied with a knot, the way Japanese furoshiki is done. The wrap becomes part of the gift.
Reusable bags and boxes
Gift bags and tins can be used again and again.
Brown paper and newspaper
Plain brown paper with string and a sprig of greenery looks smart. Old maps and newspapers work too.
The scrunch test
If wrapping paper springs back when you scrunch it, it probably has plastic or foil and cannot be recycled. Glitter and foil papers usually cannot be recycled.