Mahogany is an open-grained wood similar to oak and walnut. A smooth, flat finish requires the grain to be sealed with a wood filler prior to staining. Leaving the wood grain open to allow the character of the wood to show through and be felt by hand is another option.
Sand the mahogany with 120-grit sandpaper. Make the piece as smooth as possible and remove any pencil marks that served as witness lines as well as any tooling marks from machining.
Decide whether to close the pores of the wood's grain for a smooth finish.
Mix the wood filler per the manufacturer's instructions if you've chosen to fill the pores. Paint thinner is typically used for mixing filler to a paste-like consistency.
Press the filler into the mahogany using a nylon rag and apply against the grain. Work in sections small enough that the wood filler will not dry out before you finish applying filler.
Wipe with the grain after adding wood filler across the piece.
Allow the filler to dry for 24 hours before working the mahogany again.
Sand the piece smooth with 120-grit sandpaper, then follow with a sanding using 150-grit sandpaper.
Apply wood stain with a staining brush, foam brush or clean cotton cloth. Use caution not to flood the piece with stain, especially if the mahogany was not sealed with a wood filler. Keep a wet edge while wiping stain onto the wood.
Wait five to ten minutes before using a clean cotton cloth (different than the one used to apply stain) to wipe off excess stain.
Wipe the piece down with a tack cloth after the stain has had a chance to dry for 24 hours. Apply a protective finish such as polyurethane, varnish, shellac or tung oil.