I have been wondering about how God works in the world today. I am confused about how much we are to do before we let God take over. I believe that God wants us to do what we can and then He will do his part. Is this correct? I keep thinking about Abraham and Sarah. I don't want to make their mistake, but when do we relax and let God take over?
Let's change the nature of the question a bit because the way you phrased it implies that people have much more control over than they actually do. God is always in control and unlike people, He can decide something and have it come about just as He planned. "Remember the former things of old, for I am God, and there is no other; I am God, and there is none like Me, declaring the end from the beginning, and from ancient times things that are not yet done, saying, 'My counsel shall stand, and I will do all My pleasure,' calling a bird of prey from the east, the man who executes My counsel, from a far country. Indeed I have spoken it; I will also bring it to pass. I have purposed it; I will also do it" (Isaiah 46:9-11). At best men can only hope their plans will go roughly as they hope. "Come now, you who say, "Today or tomorrow we will go to such and such a city, spend a year there, buy and sell, and make a profit"; whereas you do not know what will happen tomorrow. For what is your life? It is even a vapor that appears for a little time and then vanishes away" (James 4:13-14).
Abraham's situation is not like your prayers. When Abraham was old, he wasn't praying for a child -- he knew he and his wife were too old. He was told by God that he would have descendants. Abraham's mistake was trying to figure out how God intended it to take place and help matters along. Instead, he should have waited for God to reveal how He planned to accomplish what He had promised.
When we pray, "Give us our daily bread," we are acknowledging that it is by God's blessings that we have food on our table to eat. But God also commands that we work. "For even when we were with you, we commanded you this: If anyone will not work, neither shall he eat" (II Thessalonians 3:10). My effort at earning money in order to buy food is a statement of how much I want my family fed. But so much is out of my control. Thus, I do what I can and trust God will see to what I can't.
Nowhere do you find the good people in the Bible asking God for something and then sitting back to wait for it to happen. You find Paul praying for open doors while he goes out and seeks opportunities to preach the Gospel. You find the Israelites praying for victory while going out to war. There are times in the past when God chooses to respond directly and tells people He is going to do something. God then tells the people what He wants them to do and then they must wait to see how God does the rest.
Therefore, when you are dealing with God's promises, you follow God's directions and wait for the result. When you are asking God for help, you do what you can and trust that God will handle the things out of your control. There are going to be times that what I think best isn't necessarily so. I might be blocked, but I know that in the long run, the best solution will appear even though I can't see it at the moment. "And we know that all things work together for good to those who love God, to those who are the called according to His purpose" (Romans 8:28).