Front End Filter for Displayed Posts

I’m not sure how to explain this, but here I go.

I’m working on a site that shows a list of posts on a certain custom category page (in this case is a list of Internship opportunities).

I want visitors to narrow down the search results by certain criteria. For example: Summer, Fall, Print, Broadcast, etc.

Now, I guess the easiest way to do that would be to add a menu of the children category for this specific category. The children being: Summer, Fall, Print, Broadcast, etc.

But I also want visitors to be able to pick (maybe by checkbox) at least a couple of criteria (for example, I want to find an internship this summer on print) – maybe even more (I want to find an internship this summer or fall on print).

Is there a way to do this? I’ve been searching online, but I’m not sure I know how to search. I’ve tried searching by “search” and “filter” and haven’t been able to find anything remotely related to my problem.

Also, I would rather to be able to use tags for this, since all the posts for that category are already tagged, but have not been categorized under sub-categories yet.