Not surprisingly, the best adhesive for Styrofoam is foamboard adhesive, special adhesive mixtures designed specifically for foamboard applications. Your manufacturer instructions will probably recommend an adhesive like these, which you can order or buy from construction retailers. Look for labels that say foamboard or foam insulation adhesive.
Construction adhesive is a more all-purpose type of glue, but it is still designed for mounting panels and substrate boards, making it an ideal adhesive mix for Styrofoam board as well. Look for a polystyrene adhesive like Handi-Stick, but read the label thoroughly to watch for any warnings against use with foam insulation products.
You may also be able to use other adhesive products for your foam board, but always take care to use versions that will not cause damage. Latex adhesives should not damage the insulation and are typically safe to use for foam boards and similar types of insulation. Liquid Nails, for example, is a latex caulk that can be used for this purpose.
If you do not want to use adhesive, or if you need to attach the Styrofoam to a surface that may not take an adhesive layer well, you can use fasteners instead. Fasteners are simple brackets that hold up Styrofoam insulation boards. They are less invasive than adhesive glues and are suitable for areas where you will not be installing other materials on top of the foam boards.