When Herod killed James and put Peter in prison, the believers in Judea prayed fervently to God for him (Acts 12.1-5). Then just before Peter was going to be brought out and presumably executed, an angel of God appeared to him and freed from the prison. It brought him back to the house of Mary, the mother of John Mark, where people were gathered and praying for Peter. The servant Rhoda heard Peter's voice as she went to see who was knocking at the gate, and she was so excited to hear him, she went running back to the group to tell them. But they didn't believe it was really him.
In many ways the Christian life is a paradox -- think, for example, of Luther's phrase that the Christian is simul justus et peccator -- and this episode provides one further illustration of this paradox. The believers are gathered and praying fervently for Peter's release, trusting that God can work even miraculous means in order to free him from the tyranny of Herod. On the other hand, when Rhoda the maid brings the news that Peter is outside, and thus that their prayers have been answered, they don't believe it. They tell her instead that she has lost her mind, or that it must be his angel -- anything but that Peter should really have been released.
In my experience, I find that I have prayed to God for wisdom or direction at various times and received answers quite soon thereafter. The difficult thing is being able to discern God's answer, which may oftentimes be subtle. It is not as if the heavens open up and a large banner written clearly in the language of your choice presents you with the directions you have been looking for. It may be small things here and there; they are from God, but you might disregard them as insignificant.
Sometimes we can both have faith and lack faith. We pray to God fervently for something terrific to happen -- say, the release of Peter from prison -- but then we cannot believe it when our prayers are answered. God preserves stories like these for us so as to increase our faith and our trust in him. And we learn thereby to be on the look-out for God in our daily lives.