I do not think you are going to see two stars orbiting about a common center of mass in a short time scale, in the same way you might see Ganymede going around Jupiter. Short period binary systems are so close together you will not separate the stars visually, but you can easily detect the varying brightness as one star passes in front of the other, if they are so oriented. Check out the Eclipsing Binary stars recommended for observation in the variable star section. W Ursa Majoris is one star I have monitored and it has a short period of about 7 hrs so if you start at the right time you can easily plot a light curve in a night.
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