Is it normal to want to be a good person without believing in God?

While I somewhat agree with you, being a kind person making one much more likely to receive kindness back isn't so much a religious concept as it is a studied, psychological fact. Approaching someone with a good attitude sharply increases your chances of being met with a good attitude, defusing a bad situation or even preventing yourself from becoming the victim of a crime (believe it or not, many criminals do have some empathy and are reluctant to kill someone who smiled, provided them eye contact and validated their existence by a nod or a wave).

Now I know that being a good person won't always save you from a terrorist attack or a drunk driver, but in a personal sense, being a good person and treating others with respect will get you much further with people unless you know how to do the asshole thing right.