The first baptisms mentioned in the Bible are those carried out by John the Baptist.
People went out to him from Jerusalem and all Judea and the whole region of the Jordan. Confessing their sins, they were baptized by him in the Jordan River. (Matthew 3:5-6)
The first baptisms into Jesus Christ, carried out after his death and resurrection, were those on the day of Pentecost when 3000 people were baptised (Acts 2:41).
However, the idea of baptism goes back to the Old Testament. For example, consider the following examples.
Noah and his family were baptised (symbolically) when they were saved in the ark:
1 Peter 3:20-21
God waited patiently in the days of Noah while the ark was being built. In it only a few people, eight in all, were saved through water,
21 and this water symbolizes baptism that now saves you also—not the removal of dirt from the body but the pledge of a clear conscience toward God.
Similarly, the people of Israel were baptized, in a sense, when they crossed the Red Sea:
1 Corinthians 10:1-2
… our ancestors were all under the cloud and that they all passed through the sea. They were all baptized into Moses in the cloud and in the sea.
Naaman was cured of leprosy through a baptismal-like process, symbolising being cleansed:
2 Kings 5:14
So he [Naaman the Syrian] went down and dipped himself in the Jordan seven times, as the man of God had told him, and his flesh was restored and became clean like that of a young boy.
Although baptism was not a requirement in the Old Testament, these examples all point forward to the baptism that would be introduced later, and that is required of all Christians (Acts 2:38).