Trust is earned

Readers have noticed that Abraham seems to be a bit distrustful regarding God's protection and promises. He tells half-truths about his relationship to Sarah to protect himself (thereby putting her at risk!), and he tries to fulfill the promise on his own by fathering a child by her handmaid. This despite the fact that he certainly has evidence for God's existence (e.g. audible voice, theophanies, revelatory dreams). 

Perhaps he'd be this way anyhow, but his insecurity may be due to the fact that Abraham was a lifelong pagan before God summoned him from Ur. Pagan gods are fickle and faithless. You can't count on them to keep their promises–assuming they even deign to make promises. Pagan gods betray their fellow gods. 

So it may be that Abraham had to overcome his inbred suspicion regarding Yahweh's bona fides. He never met a God like this before! He had to unlearn what he expected from gods. 

We're so used to the Bible that we can miss the revolutionary nature of biblical theism. Take the creation account. A universal God who preexists creation. Paganism didn't have a universal God, but many territorial gods. Likewise, in paganism, the world preexisted the gods. And the world might well outlast the gods. Try to read the Bible through pagan eyes, and see how revolutionary it would be to ancient Near Easterners.