A map like this is as complete as you'll find for Middle-earth. Tolkien's stories are heavily rooted in this northwestern corner of the continent. I feel like the south and east primarily existed to give the world depth. He wrote little on those lands because they were not of much concern to the main story. It's difficult to guess if he would have ever gotten around to writing much more about them than he did.
Sauron, like Melkor/Morgoth before him, seemed interested in ruling whatever he could, but more attracted to things he would have trouble controlling. Such is greed and the desire to dominate - always wanting what you can't have!
The Haradrim (south) and Easterlings (east) were evil and combative; I get the idea of barbaric as well. They fought alongside Sauron, whether under his direct influence or just up for supporting something they can wage war with. Thousands of years before the War of the Ring in the Third Age, Melkor had already ventured south and east, and we know that the men there were heavily involved in "Melkor-worship," as Tolkien calls it once. Sauron too had already had an influence there years earlier as we can see in a ringwraith's identity: Khamûl the Easterling,second greatest of the wraiths.
Also, we know upon arrival of the Istari (wizards), Saruman and the two blue wizards went into the east and only Saruman eventually returned. This time spent east could've been part of Saruman's corruption and betrayal. (While Tolkien was never completely decided about the fate of the two blue wizards, he said they stayed in the east for a long time, speculating that they began cults or magic traditions of their own.)
You should read
The Silmarillion as you'll find an Easterling named Ulfang has a part to play. Don't let us spoil it for you.
