I love this question! Discovering and utilizing your spiritual gifts is one of the most exciting adventures a person can have with God. The Bible says spiritual gifts are abilities God bestows on every believer for the common good of the body of Christ. They're a large part of the answer to the question, "What should I do with the life God gave me?"
Passages like 1 Corinthians 12, Romans 12, and 1 Peter 4 go into specific detail about what these gifts are and how they should be used. It's clear every spiritual gift is a reflection of God's nature that you carry within you. And because your spiritual gift reflects God's design and direction for your life, you'll find great passion, joy, and satisfaction in expressing it. Your spiritual gift also will be a place of deep spiritual formation in your life, as God uses it both to powerfully connect you to him and to expose areas of your soul that need his forgiveness and redemption.
So here's a process to start:
Pay attention. Notice the things that energize you and seem to come naturally. Remember the quote from the movie Chariots of Fire when Eric Liddell explained to his sister why he was postponing his return to the mission field in order to race in the Olympics? "Because when I run, I feel the pleasure of God."
Every spiritual gift gives off clues. Your spiritual gift will cause you to react a certain way in a given situation. If there's a problem, people with the spiritual gift of shepherding will be immediately concerned that people are cared for and growing in Christlikeness as a result of the issue. Those with the gift of intercession (prayer) will immediately say, "We need to pray about this," while those with a leadership gift will begin looking at solutions for the problem.
Try. Once you've gathered enough information to create a list of some possible gifts (perhaps gifts of mercy, evangelism, encouragement, or hospitality), exercise your options. A great place to start would be a volunteer position at your church. While you're trying it out, you'll start to discern whether you're good at it or not. Also, others will tell you!
When my kids were young, our church needed help in the nursery during the worship services. I volunteered for a three-month opening. I didn't feel the pleasure of God; the children didn't feel the pleasure of God. It was so not my spiritual gift. Part of learning what you're good at is having to go through the pain of learning what you're not good at.