Find corroboration. A hunch or suspicion may pique the reporter's interest, but it's critical that you build enough evidence to support the story.
Interview sources. Usually this starts with people on the periphery of the investigation who can provide background information. These interviews will provide threads and other paths you can follow as your investigation deepens. As you gather more information and research, you can target the main players and have these interviews on the record. Keep the reason for your interview vague when you schedule it so that they don't have time to craft a defense; don't lie about your motives, though.
It's best to confront people suspected of misdeeds when you have all the evidence you need. If they won't grant you an interview, wait outside their office or in the parking lot until the person comes out. When she still doesn't respond, it shows the public you made an honest effort for comment.
Make public records requests. Each states have varying sunshine laws, but most require any documents that originate in the public sector be provided to the person who requests them. This can be especially useful in the case of damning emails sent between elected officials, but also shed light on dubious transactions.
Visit the scene of alleged misdeeds. If it's the scene of a neighborhood contaminated with cancerous toxins, for instance, walk the streets and talk with people in the neighborhood. They can shed more light on their woes and a tour of the site will give color to your story.
Consult frequently with an editor or more experienced reporter. Having a guiding voice for your story can keep you from getting discouraged and help you think of other unexplored angles.
Vet the story with a lawyer for possible libelous statements if you believe the story is especially inflammatory. The people exposed will not like it and probably attack your paper and writing, so it's best to have a defense prepared in advance and any legal footholds in their favor eliminated.