How can I change color of lampshade that still lets light through?

Paint it! There are many fabric paints on the market. Or recover it in another fabric.

Hmm I would think a dye would probably be the safest bet to make sure light would still shine through etc. I tried painting a lamp shade once but I just couldn't get it even enough for it to look solid (and not splotchy) when the light was one. Maybe a fabric dye in a big tub of water so you could dunk it?

A nice ribbon, fringe or cording that matches the room. Glue around the top and the bottom of the shade.

As mentioned, you could paint or dye it. You could add beads, fringe, decoupage paper napkins (with lovely pictures) onto it, or make your own lampshade Out of parchment, ricepaper, fabric, wide strips of ribbon or lace...oops! 😲 I could go on and on. Think of a white shade (or anything a blah white) to be a canvas where you can let your imagination run wild! 😁

Fabric or spray paint

I crocheted a cover for it using black thread.