Are professors allowed to participate in political activities?
Since you are concerned, it must be a good idea to look into the rules, as @Penguin_Knight suggests.  But it would be absolutely shocking if you were not allowed to volunteer for a particular candidate or party: by restricting you from doing that, the university would be infringing upon your own political freedom.  


  I am concerned that I am in a position of power, and thus if I were to be out campaigning for a particular candidate, this could affect my students' freedom to vote, should I run into them while I am volunteering.


I'm honestly not sure what problematic activities you're envisioning here.  A university professor is allowed to be a real boy or girl.  Students may run into you while you're outside doing real person stuff, and they have no right to being "protected from you" any more than anyone else they might encounter.  How does running into a student affect that student's "freedom to vote"?  


  Are there any rules in the university that forbids professors from participating in political activities? How much is acceptable? Can I help out with phoning people to get them to register to vote? Can I knock on doors in person? Could I run for office eventually? 


Again, the answer to the first question is that if you want to know the rules of your university, you should not ask us, because you know which university is yours and we do not.  (Added: As Tom Church points out, there is a small but positive number of US universities for which the answer will be very different from the general case!)  The answer to the last three questions is that I would certainly expect you to be able to do all three of those things (again, you're a real person; you have the same right to knock on people's doors as anyone else!). If you successfully attain office, then this might interfere with your academic schedule and commitments, but it would be up to you to resolve that: e.g. if you live in a small town and get elected to a town council that only meets on Thursday mornings...probably okay.

If you feel strongly that (i) you want to be openly political and (ii) that you do not want your job to be even slightly at risk, here is some advice for that: I suggest you keep a clear separation of your political and educational activities.  When you bring up politics in the context of a course or student supervisory relationship, it should be in the service of an intellectual/academic point you are trying to make, not your own political activity.  When political ideas come up naturally in coursework, you should make an effort not to signal or imply that your own political beliefs are "the right answer".  You should certainly not try to politically proselytize with your own students or suggest that their grades will suffer if their political views do not align with yours.