How should the believer direct his or her prayers?

Everything

Prayer should always be made for things that are important. However, we should remember that nothing is too small for God. Basically we can pray for anything great or small.

Do not be anxious about anything, but in everything, by prayer and petition, with thanksgiving, present your requests to God (Philippians 4:6).

Jesus' Example

We can get an idea of what sort of things to pray for by observing the prayers recorded in the Bible.  For example, Jesus prayed for workers to be sent into the harvest.

Then He said to His disciples, The harvest is plentiful but the workers are few. Ask the Lord of the harvest, therefore, to send out workers into His harvest field (Matthew 9:37,38).

Others

Our prayers should also contain requests for others. The Apostle Paul encouraged us to think of others.

Do not merely look out for your own personal interests, but also for the interests of others (Philippians 2:4 ).

Make A List

A practical thing to do is to make a list of those for whom you are praying. Among other things this list should include: the unsaved, people who are sick, those who are in the ministry, friends and relatives.

Summary

We should not be only be looking after our own interests but also the interest of others. Our prayers should reflect that.