Two washes has worked well for me. To keep chemistry to a minimum, I have also pinned fabric to a board and then sized it with two coats of a 3% gelatine solution. The fabric goes stiff and can be handled just like paper. once the Cyanotype is made, a short soak and then rinses in warm water will dissolve and remove the gelatine. It is worth a few tests as the trick is to get enough gelatine to stop the chemistry soaking right into the fibres, but not too much that the image is only in the gelatine and not the cloth and then all washes out.
The closer the weave of the cloth, the easier it is to capture details in the print. I have done some printing onto linen which was quite coarse and the weave of the fabric becomes very prominent to the eye. As the 'threads' get thicker, they also seem more prone to changing their shape and so fine details that are smaller than the thread size tend to get a bit distorted and lost.