You need to take a course and then have to have a place to trap. I took the course 3 years a go and I am currently trapping private land and I am on the list for MTO contract work. You need 10 years plus experience to have any chance of getting your own Crown land trapline, unless you find someone looking for a helper or have a relative with a line.
I managed to sign up a bunch of private land and I also due nuisance beaver removals via word of mouth and a small ad in the local newletter. So far I have 25 beaver in the freezer from over the last 2 weeks, all from within 10 km of my house. Once the ponds freeze I will move onto marten, fisher, lynx, fox and coyotes.
The link below also lists the contact info for all the MNR District Office's, give the closest one a call and they may be able to tell you when the next course is or they will put you in touch with the local trapping association.
Best of luck.
http://www.furmanagers.com/viewsection.aspx?section=licensing
Licensing
Applying for a trapping licence
Persons applying for their first trapping licence, or those who have not held a trapping licence the past five years, are required to complete the Fur Harvest, Fur Management and Conservation Course. These courses are given by Ministry licensed instructors, and students are required to purchase the Fur Harvest, Fur Management and Conservation Course manual. Following the course, each student will be subjected to a written and practical field examination. Individuals wishing to take the course should contact the Ontario Fur Managers Federation or local MNR district/area office. Individuals who are already licensed as trappers but, have not taken the course are encouraged to do so as are persons who for various reasons are interested in furbearing mammals and their management but do not intend to apply for a trapping licence.
Obtaining trapping licences and registered traplines
Applications for new trapping licences are authorized by the Ministry as vacancies occur, and according to set procedures. The successful completion of the Fur Harvest, Fur Management and Conservation Course does not guarantee a person a trapping licence. For information, please contact the Ontario Fur Managers Federation or a District/Area office of the Ministry of Natural Resources located in the area where you wish to trap.