Can I text someone across the country

If you have text messages through your cellular carrier, yes.

If the person has a iphone, you have an internet connection-you can iMessage them.

Basically, yes. The only potential cost would be for the Internet service you are using. For example, if this is your home's Internet connection, they you would be paying your ISP for the "privilege" to access the Internet ... so, technically, it would not be "free." On the the other hand, if you are using someone else's Internet service, then there shouldn't be any costs involved.

As the others, I also communicate with friends overseas using either FaceTime or iMessages. So far, this has come at very little or no cost at all to me.

I routinely text across the Atlantic Ocean, and occasionally all the way to Australia.

Well, it’s not clear whether “texting” means using your cellular carrier’s SMS or Apple’s iMessage.

What is the difference between iMessage and SMS/MMS? - Apple Support

In either case, texting across the country should be the same as texting across the street.