If you ask at the fabric store, there are usually free handouts explaining the different types of interfacings and stabilizers available. Depending on your choice of hand or machine quilting, and the amount of stiffness you want, you can use different weights. Remember that the shiny or dotted side is the glue, so keep it facing AWAY from the iron.
Tear away and sew-in stabilizers are useful in other ways, but not what you need right now.
My guess, from what you've said, is the fabric is really too thin in its present form to be good for quilting. Even with interfacing, after years of washing, it will break down faster than regular weight fabric. You may be happier with another choice for sashing.