No, the Old Testament isn't out-of-date or irrelevant to us today. It is just as much God's Word as the New Testament, and God has much to teach us through its pages. The Bible says, "All scripture is given by inspiration of God, and is profitable" (2 Timothy 3:16, KJV).
Let me suggest two "keys" to help you unlock its treasures. First, learn from the people you meet in its pages. The Old Testament tells of some of God's greatest servants -- Joseph, Moses, Jeremiah, etc. -- and God has much to teach us from their example. But it also tells of people who failed God. Why did it happen? What were the results? They too can teach us, warning us of what happens when we turn aside from following Christ. The Bible says, "These things occurred as examples to keep us from setting our hearts on evil things as they did" (1 Corinthians 10:6).
The second suggestion is to begin with the parts of the Old Testament that are easiest to understand. The book of Psalms, for example, is the "hymnbook" of the Bible, while Proverbs gives us practical lessons for daily living. Make them part of your spiritual "diet" every day.
At the same time, don't neglect the New Testament. The Old Testament foretells Christ's coming; the New Testament reveals His coming. The whole Bible points to Him, and He is its center. Is He the center of your life?