Facebook: Pages or Groups?
Well, the answer is: use both! Like spoons and forks, or shampoo and conditioner, each are good for different purposes, and work best together.
Pages are for marketing. They focus on ease of connecting with your public audience. With one click, a Facebook user can both express their like for your organization, and subscribe to updates. Pages should be public, otherwise you're not quite using them to their full capability.
Groups are for discussion. They focus on building a community of people who talk about the station, and allow you to make internal announcements to this community. They work best if they include just the volunteers directly involved in the station itself. Groups should be closed, but not secret. Then, people can find them easily if they want to join, but you can still keep a handle on who joins, which prevents spam and poor-taste discussion.
However, if you are trying to use Facebook for contacting contacting interested folk (i.e. potential donors), then Facebook is actually not the best way. Instead, alongside Facebook, use a platform for nonprofit donor management (see below) to really get the most out your fundraising and outreach efforts.
More details about Pages and Groups
Although your posts in both Pages and Groups can show up in the Facebook timelines of the people who have liked or joined them, you should use Pages if visibility itself is what you really want.
In other words, Pages are great marketing tools, but not good for discussion, since they are tailored for broadcasting posts to a potentially unlimited number of Facebook users who freely subscribe by liking the page. Groups are great for discussion, but very restricted as marketing tools, since you need to (or rather, should) approve who joins them.
So it's best to have both, and learn how they work in tandem. The best way to learn this is to just to set up both and start using them.
How do I connect with listeners and underwriters?
In a small station's early stages, to connect with those local people who are interested in your station -- especially potential underwriters -- here is an idea:
Create both a Page and an open Group, and on your page, advertise your group, and vice versa. Then you have a better chance of connecting to people (through the group), but you are also building your a Facebook subscriber base for marketing (through your page).
Later, when you have things running better (more people, and things are getting more clearly defined), convert it to a closed group, and really start pushing the page for marketing, versus the group for discussion with volunteers rather than volunteers, rather than listeners and underwriters (which should be handled by now with a member database instead, see below...).
Member databases: You should use these, too.
At the same time, the best tool for building a donor and underwriter funding base is actually not through Facebook -- it's through email newsletters and a using member database platform. Then you can have contacts and emails, with notes on each person (esp. what stage they are in their interest in donating or underwriting). For this, I'd look into Saleforce for Nonprofits, which is free if you are a 501: http://www.salesforce.org/nonprofit/fundraise/
(also no I don't work for Saleforce -- and as a speculative aside, I guess they're just so big at this point that they can create a whole nonprofit platform as a giant tax write-off or something?)