Will my insurance rates go up if I rear end someone and only my car is damaged and I make a claim to my insurance company to fix it?

Will my insurance rates go up if I rear end someone and only my car is damaged and I make a claim to my insurance company to fix it?
Very likely yes. This is especially true since you are at-fault for the accident. (meaning you caused the accident).

Some insurance companies reward loyalty and may have "accident forgiveness" program if you have been a long time customer with them.