Don't believe or disbelieve when investigating
It is easy to say but hard to do. Believers will inevitably see a seemingly good case as the proof they want. Hardened sceptics, meanwhile, may make only a superficial investigation because they already 'know' it can't be paranormal as it doesn't exist! Neither approach is at all helpful.
The best way to approach any paranormal investigation is to simply see what is there and what is not. The main method of investigation is to try to eliminate natural causes for the reported phenomena. The idea is that whatever is left (as Sherlock Holmes didn't quite say), however improbable, must be the paranormal.
One problem, though, is how do you decide that you've eliminated ALL possible natural causes? There may be some that you haven't considered or that are actually unknown to current science! This is a way of seeing where you are on the belief / sceptic scale. A believer may eliminate a few obvious natural causes for a phenomenon and announce in triumph it is proof of the paranormal. A sceptics may exhaust all possible natural explanations but still 'know' there's something he's missed!
One of the problems we have in the field of paranormal research is deciding how to make sense of all the evidence. A lot of it seems contradictory and some cases are suspiciously unique. I think the reason for that is that many cases are investigated in a biassed way producing results that support their investigator's prejudices.
If we remove such poorly investigated cases there is still a core of reasonably consistent evidence. For instance, most hauntings consist of a few odd noises, strange lights and smells and some object movement (I should explain that this object movement is not usually witnessed, it is merely noticed after the event!). This core certainly doesn't appear quite as inexplicable as the wilder stuff that passes for evidence in much ghost literature. There IS something there to be explained but not as much as you'd think!