How much activity should a person with superficial phlebitis be engaging in? How much should they be resting?

As tolerated. Oftentimes phlebitis is painful enough that being on your feet is intolerable. If that is the case, then lie down and put your feet up. A soon as your leg feels good enough to bear weight, you should be as active as your discomfort level will allow you to be.
As much as tolerated. If you can tolerate walking and staying active this may help speed resolution of your phlebitis and more importantly prevent extention of the superficial clot into a deep(major) vein. Very strenuous activity can increase your discomfort and should be limited until you feel better.
Last thing you want. Is to sit around. Move as much as pain allows and keep leg up while sitting.