1. Jesus kills a tree because it didn't have any fruit on it for Him to eat when He was hungry. It's a bit disturbing to see Jesus lashing out like this and killing something. It does give us a glimpse toward the judgment, though. When Jesus comes again, He will judge people for not doing what He wanted them to do, just as He judged the fig tree for not producing fruit like He wanted it to do. As the Creator God, Jesus has the right to demand that His will be followed in all of creation, and to judge anything that does not follow His perfect, sovereign will.
2. Jesus tells the disciples that they can accomplish anything if they have enough faith. This is another disturbing aspect of the story. How many people have ever moved mountains? How do you get the kind of faith Jesus was talking about?
Faith comes from God. It is a spiritual gift. Christians should also be guided by the Holy Spirit in what they are praying for. If God tells you something is going to happen and you pray for that thing to happen, then you are putting your faith in what God has said, and because God wills it, the thing will be accomplished.
Take the moving mountain, for example. I would never randomly decide to rearrange the landscape to such a degree on my own initiative. But if God communicated to me that He was going to move a mountain, and I prayed for that mountain to move, then my prayer will have been answered because I put my faith in what God said and prayed accordingly.
The key to having your prayers answered is to get in tune with what God is doing or about to do and then pray accordingly. If you do so, you will be privileged to have known about the event ahead of time and to have seen it happen.