Question
My wife was given the orchid as a gift and it has since
flowered. It said on the instructions that it should be
discarded after flowering. Flowering is now over but on the
stem between the top of the soil and the first leaf several
whitish spikes are growing. We wondered what these were and  
what action we could take to continue growing the orchid.

Answer
Russell, it is likely that the whitish spikes are new roots.  You should locate a source of orchid potting mix at a garden center.  While there purchase a plastic pot with lots of drainage holes in the bottom  (Lowes or Home Depot usually carry orchid potting mix at their garden centers).  Soak a portion of the orchid potting mix while you unpot your plant.  Rinse the old potting mix from the plants roots after unpotting the plant and remove any roots that are mushy (healthy roots are firm).  Then place the plants roots in the clean pot.  You may rotate the pot while you lower the root ball into the pot.  This will allow the roots to distribute themselves against the wall os the pot and minimize damage to them.  Then, add potting mix, tapping the pot to permit settling about the roots.  Continue this until the potting mix is about 2 inches from the top of the pot.  Place in an area where the pot can drain off the excess moisture.  When it stops draining, place the plant into its growing area.  They prefer an east window, but a west or south window will also work.  Store the dry, unused orchid potting mix as you should repot the plant after each flowering.  It should flower again in bout 6 months if it remains healthy.