There are three basic factors that affect the nature of the Savannah cat behavior, lineage, generation and socialization. These three factors follow the nature vs nurture argument with nature being breed lines combined with generation and nurture being social up bringing. The Savannah breed development is still at it's infancy (2014) and most Savannah cats have a very broad range of behaviors.
If a breed line has a tendency for a specific behavior over other behaviors it is likely to be passed to the breed lines offspring. As outside lines are used there is a merging affect of the base behaviors.
When breeding lines starting from early generations such as first filial and second filial generations (F1 and F2 savannahs), behavior stemming from the wild out cross, the Serval, is more apparent. Behaviors like jumping, fight or flight instincts, dominance, nurturing behaviors are more noticeable in early generations. Since fertile males that are F5 and F6 are used in most breeding programs later generation Savannah cats behaviors tend to act more like traditional domestic cats. Over lying behavior traits for all generations are high activity and high curiosity.
Probably the most influential factor is early socialization. Kittens socialized with human contact from birth and human interaction each day reinforces kitten and cat human interaction behavior that lasts through out the cats life span. Kittens within litters will tend to have varied social skills with some that like human interaction and others that fear it. If kittens that fear humans never grow past that fear they will tend to exhibit a more shy behavior and are likely to hide went strangers are present. Kittens that look forward to their human visits and like to engage in play with their humans tend to grow to cats that are more welcoming of strangers and less frightened of new environments. These cats tend to be more the life of the party versus a cat that will find a hiding place until the party is over. Human cat socialization should be practiced each day with positive reinforcement for kitten to grow into a well rounded social Savannah cat. Kittens that go for long periods of time without human interaction and only interact with their mothers or siblings many times do not develop a strong bond with humans and tend to be less trusting of their humans. These kittens tend to be more shy and are likely to hide when humans that they don't know are present.