You are right; sometimes our prayers can seem very self-centered. When they are, it's usually because we're only interested in getting God to do something for us, instead of seeking His will, whatever it is.
But this doesn't mean it's necessarily wrong to bring our needs and desires to God in prayer - not at all. God loves us, and He wants us to bring our needs to Him and trust Him for the outcome. The model prayer that Jesus gave His disciples includes this petition: "Give us this day our daily bread" (Matthew 6:11, KJV). But that isn't a selfish prayer; it is simply an acknowledgement that we are dependent on God for everything - including our food.
Thank God for the privilege of prayer - a privilege He has given us because Jesus Christ opened heaven's door for us through His death on the cross. Thank Him too that you can bring everything to Him in prayer - every concern, every need, every desire. The Bible says, "Do not be anxious about anything, but in everything, by prayer and petition, with thanksgiving, present your requests to God" (Philippians 4:6).
At the same time, make sure you aren't just praying for yourself. Pray for others, and pray for God's work in your church and in other parts of the world. It's one of the most important things any of us will ever do.