When you first begin your starter you want to keep it at room temperature and feed it daily (twice daily is even better).  Do this for a couple of weeks to get a nice healthy starter going.  After that, if you are using the starter most days then leave it out at room temp and feed daily.  Otherwise, refrigerate and feed weekly.  If you miss some feedings during refrigeration, no worries, a healthy starter can often be brought back from the brink of death with a bit of tender loving feeding.  I bring back to room temp and feed my starter the night before I plan on using it.  You do want to keep it covered to keep the nasties out.  I keep mine in a old mason jar covered with a sandwich bag held in place by a rubberband.  A healthy starter can created a fair amount of gas after it is fed so I wouldn't use anything super airtight.