I think two major feelings expressed in Bradbury's story are disgust and anger. I believe that readers are meant to be disgusted by Eckels's pompous, selfish, and cavalier attitude. Throughout the first half of the story, it is obvious that Eckels doesn't care about anybody or anything other than himself and his dinosaur.
"All you got to worry about is—"
"Shooting my dinosaur," Eckels finished it for him.
Even once Eckels is back in the past, he still acts as if everything is a game.
Eckels, balanced on the narrow Path, aimed his rifle playfully.
The reader's disgust for Eckels deepens when he can't support all of his bold bravado. He is terrified at the sight of the dinosaur to the point where he can't even follow the pathway back to the time machine.
I also believe that Bradbury wants his readers to feel concerned about their actions. Bradbury's story illustrates the concept of the "butterfly effect." It basically says that small changes or actions can have large consequences. Bradbury wraps a neat little story around that concept, but I believe that Bradbury would like his readers to feel concerned about their own actions toward other people, technology, Earth's resource management, etc.
There are at least two distinct feelings that are meant to be instilled by the story; disgust and fearful awe.
The description of the tyrannosaur strongly emphasizes the size, danger, and repulsive aspects of the creature. Bradbury is attempting to convey that the animal is huge and offensive in a way that mere words have difficulty conveying, that the animal is triggering a primal instinct in Eckels that goes beyond rational thought.
Additionally, we are meant to feel disgust at Eckels' cowardice, and a sort of sickened pity at the way he conducts himself afterward, and the hostility that Travis displays toward him. Finally, we are meant to feel disgusted or shocked at the changes in the world that Eckels' mistakes have caused, being portrayed as objectively bad changes that cannot be undone.