Is Android messaging like iMessages? Does Android have free texting like the iPhone? If I'm texting to another Android user, will it be free like it is for iPhone users, or will I have to pay for the messages? I'm new to the whole Android experience, and really need some help. And I don't want to use a 3rd party app like WhatsApp to text the other Android users.
No. On all Android phones I'm aware of, the "Messaging" or SMS app is just an SMS app. It doesn't try to sucker you into using a proprietary chat service by pretending it's giving you free texts.

There are many IM and chat services for Android if you want to use one instead of texting. Like the iPhone, there's even one that ships with the phone, but unlike the iPhone, it's a separate app, called Hangouts. Hangouts looks and feels like a chat system, not an SMS app, and it includes voice and video calling, sending files, and group calling.

Android allows third-party apps access to all the phone functions, so if you really want to use SMS and a chat system in the same app, I'm sure you can find a third-party SMS app that includes both, such as Skype.