I would like to know the best way to get into the field of demonology. Where and how should get started?
I know that demonology is alive and well in many fundamentalist Christian churches. (The History channel regularly features ministers of such churches on their many shows about heaven and hell.) I would imagine that such churches have apprenticeships of some kind, but I wouldn't expect that they have anything as structured as the Roman Catholic programs of study, as discussed in the post by herappleness.
You may try some internet searches using terms such as "demonology evangelical" or "spiritual warfare" and make note of the names and institutions that are identified. Try contacting one or more of them to see if they might be better able to answer your question.
From my point of view, there are no demons. That is, there are no supernatural evil beings with an actual existence. Now I know, as cited above, some religions do believe in demons and devils and supenatural possessions and the like, and you could enter the world of theology to study that kind of thing through some school of religion, but I don't know if that's what you're after.
What are you after? Do you believe in demons? Have you seen or experienced any yourself?
Maybe you're talking about personal demons: bad and recurring thoughts and habits and addictions and ways of behaving that torture your mind and everyday existence and the people you know and interact with. If those are the kinds of demons you mean, then the study of psychology might be the way to go, or you might want to see a therapist to get at those inner "demons."
The Catholic Church is the only place I know of that confers an actual title of "Religious Demonologist", for which you actually have to so some hard studying and complete a dissertation.
A regular demonologist, according to the information I got, can become one either through theology studies (the branch of Theology has a myriad of sub-branches many of which deal with history, spirituality, psychology, counseling, pastoral, etc), or some even just declare themselves to be demonologists, read two or three books, join a paranormal group, and go to business. This, however, is a very dangerous step to take.
A real demonologist is a researcher and has to not only get the facts quite straight about the process and historical context of each case, but you will also have to sit with the client and determine whether the person is psychologically sound of mind. You must also be able to admit that, just as you believe in demons, you also believe in their counterparts as philosophy and theology are balancing systems of thought. For this, you may want to obtain additional certifications to ensure that your reputation is solid, and for your practice to be successful it its own unique way.