You generally have two different options as a first boss: King Slime or Eye of Cthulhu.
Both can spawn automatically. King Slime spawns rarely on the outside of the world or at the end of a certain early-game event, and Eye of Cthulhu can spawn at the start of a night if you have 200 HP and at least four NPCs.
If you want to streamline them spawning, you can craft a summon. Boss summons in the early game are crafted at demon altars, which are in the Corruption or Crimson caverns. The King Slime one is made of a Crown (some Gold or Platinum and a Ruby) and 20 gel, and the Eye of Cthulhu one is made from 6 lenses. You can also occasionally find Eye of Cthulhu summons in chests underground.