Anne, yes the ink will dry on the gesso. You may want to heat set it to speed up the drying time. You just need to cover the entire surface and it doesn't have to be super thick. The Gesso soaks into the canvas so the ink sits on top instead of soaking into the canvas.
You can use any number of mediums on top of Gesso: ink, acrylic/oil paints, mists, mod podge, etc. What I mainly use are acrylic paints to get the background colors I want. I buy the cheap bottles of paint because in the long run it's less expensive than using ink. You might want to consider buying white acrylic paint and then color with reinkers to stay in the SU color palette. If you don't want the colors to blend, heat set between each color.
Here on SCS there is a mixed media challenge - you check it out to get some ideas. Mixed media is addicting but so much fun.