Visit your local fabric or craft and hobby store. There is a vast array of washable fabric paints to choose from. Buy a few small bottles of different brands and give them a try. Then you will find your favorite brand.
I would think that a dye would be better than a paint.
Here is a project that might help
Buy fabric or printing inks.
You can stencil or use rubber stamps using the fabric paints.
You can stamp an image, then paint over.
You can also trace or you can use transfers.
Printing is a little more difficult.
You can make an image by drawing onto a polystyrene tile. Or do a linocut image. Put a small amount of printing ink/paint, onto a tile or glass from a frame. Then use a brayer( rubber printing roller) to roll out the paint until sticky texture. Then use your image tile or lino cut face down onto the ink. Roll over with another brayer or old rolling pin.
Then lift and use to print onto your napkins. Best test print first onto paper. Once dry. Set with an iron.
I'm Linda. There are a lot of designs that you can paint onto cloth napkins using fabric paint. You can use a stencil for a design, tape to make straight painted lines, stamps either purchased or homemade to stamp on designs, a glass or clean tin can to stamp on circles or you can tie-dye or design your own any way you like. Here's a tutorial for one way to paint on cloth napkins that should get you started. Wishing you the best!