Here's a simple card idea that uses standard 1/8 inch quilling strips as the weaving material. If you prefer to cut your own paper, you'll need a straight edge, cutting mat, and craft knife... or even easier, run sheets of colored paper through a narrow cut shredder.
To weave, place a piece of clear acrylic or waxed paper over a sheet of 1/8 inch graph paper.
Begin by laying strips vertically, using the lines as a guide and skipping each space for an open-weave look. Small pieces of tape can hold the ends in place, or place a piece of foam core board under the graph paper, and stick a straight pin in the end of each strip to keep it from shifting. This will save time fooling with fiddly bits of tape.
Do standard over and under potholder style weaving with horizontal strips... easy peasy!
Occasionally apply a tiny bit of glue with a T-pin or toothpick where the weaving intersects to give the grid some support. When you've woven a rectangle that's approximately 3 x 5 inches, rule the edges in pencil and trim the end of each strip on a diagonal. Dab a glue stick on the back of the weaving and then press the woven grid in place on the center of the card front.
My card has a layer of orange vellum overtop cream cardstock, but that step really isn't necessary unless you like vellum as much as I do... just start with whatever color cardstock best coordinates with your embellishments.
I crossed the weaving with a horizontal band of color-coordinated ribbon to add interest and glued metallic charms on one inch cardstock squares. These were centered, turned on the diagonal, and glued in place.