To get the sharpest curves, use a hand coping saw. Always drive screws or brads through the hardboard into the material below, never the other way. If nailing, use the special hardboard brads that have very small heads. The cut edge should be smoothed with a medium, then fine grade wet or dry sandpaper or aluminum oxide paper, or even silicon carbide paper. Wrap the abrasive paper around a sanding block.
All wood working adhesives work well with untempered hardboard.
When gluing the smooth surface of the hardboard, roughen the surface with coarse sandpaper. When working with tempered hardboard, sand the area and use epoxy glues.