God has given us animals as a source of food.
The fear of you and the dread of you shall be upon every beast of the earth and upon every bird of the heavens, upon everything that creeps on the ground and all the fish of the sea. Into your hand they are delivered. Every moving thing that lives shall be food for you. And as I gave you the green plants, I give you everything. (Genesis 9:2-3)
As God has given them to us — it can’t then be a sin to hunt them for food. Jesus himself was involved in fishing with the disciples (for example John 21:4-14).
Is the causing of hurt always synonymous with sin? God and Jesus discipline and directs us through the events of life — some of which can hurt us. Such events are always for our ultimate good (see Hebrews 12:5-11, Romans 8:28). Sometimes we have to do or say things to friends that hurt in some way, but we have their best interests at heart (Proverbs 27:6). Being hurt is not always caused by sin.
Is it acceptable to hunt and fish solely for pleasure? The Bible doesn’t directly comment on such actions, so it is a matter for our personal conscience. I personally think it is unacceptable to hunt or fish if we are doing it purely to inflict harm and destroy creatures that are part of God’s good creation — which he cares for (Genesis 1:21, 25, Psalm 50:10-11, Psalm 147:9, Matthew 6:26). Interestingly, two of the Bible’s most well-known hunters are remembered in a negative light (Nimrod: Genesis 10:9, 11:4 and Esau: Genesis 25:27, 33).
Is pest control acceptable? God recognises the problem that large numbers of unwanted animals pose (Deuteronomy 7:22). We could also suggest that it is acceptable to hunt or fish creatures that are deemed as pests.
Humanity has been given dominion over the animals (Genesis 1:26, Psalm 8:6-8). This involves a duty of care:
Whoever is righteous has regard for the life of his beast, but the mercy of the wicked is cruel. (Proverbs 12:10)