My dog bit me, cleaned the cut with water and used neosporn, does that mean it'll be ok?

Probably BUT. Most dog bites do well with simple measures like you describe, a thorough washing and dressing, BUT the area involved and the severity of the bite sometimes requires further care. Also you should be sure you have had a tetanus immunization in the last 10 years and We always make sure the dog has been vaccinated for rabies (since its yours, you should know).
Maybe. Animals particular cats can cause bad infections when bit. Dogs less likely but after the wounds are cleaned I recommend updating tetanus if > 10 years and just making sure the wounds do not need antibiotics. Please get checked.