The description of the free skill, Conceal, seems to imply that it only works when hiding a person if they are stationary. Also, there doesn't appear to be a different stealth-based skill that would cover sneaking around. Am I missing something here, or do characters need to take a specific, different skill to be stealthy when moving around?
You're not missing anything. Conceal does not give you the ability to sneak. You would need to take a do-it-yourself (ie, not-free) skill to do so.
Note: the section "Do-It-Yourself Speed Skills" includes "Move Silently" which may be of use to you.