The path will lead you to a larger area with more Squirts and some tougher enemies, the Stinkeyes. Stinkeyes are slow but can deal significant damage if they get close. Use your Breaker's Bow to take them out from a distance, or get in close with the Cael Hammer and dodge their attacks. Remember to pick up any Fragments dropped by defeated enemies; these are your currency.
You'll eventually reach a glowing pedestal. Interact with it (E key) to retrieve the first Shard. With the Shard in hand, a portal will open, leading you back to the Bastion. This marks the completion of your first mission. Back at the Bastion, Rucks will explain the purpose of the Shard and the Bastion itself. You can now access the Arsenal to upgrade your weapons.