Avoiding massive controller: where to put individual files

core is to extend or replace core libraries.
model is for your data model (database classes).
libraries is for your other classes.
helpers is for file containing functions.
third_party is for files from other project used by you application.

Perhaps you could be more specific? Like perhaps delineate where core libraries differ from non-core libraries?

I should have say "core classes". CodeIgniter's core classes are in system/core

http://www.codeigniter.com/user_guide/ge...asses.html

I've heard that model files may do more than just deal directly with a database. Do helpers contain ONLY functions? Might they not contain classes?

That's the way CodeIgniter's is structured. Nothing prevents you from putting whatever you want in those files. In my models I also have functions that do other stuff than database operations. Like data validation, for example. Stuff related to the database model.

I suggest you read the User Guide to learn the basics of CI :

"Models are PHP classes that are designed to work with information in your database. For example, let's say you use CodeIgniter to manage a blog. You might have a model class that contains functions to insert, update, and retrieve your blog data."
http://www.codeigniter.com/user_guide/ge...odels.html

"Helpers, as the name suggests, help you with tasks. Each helper file is simply a collection of functions in a particular category. "
http://www.codeigniter.com/user_guide/ge...lpers.html