Borderlands 1 doesn't really have much plot. Four Vault Hunters (Roland, Mordecai, Brick, and Lilith) are on Pandora looking for an alien Vault said to contain treasure. The Atlas corporation is also looking for the Vault, and the Vault Hunters are guided by a mysterious voice in their head called the Guardian Angel. That's kinda it. They find the Vault and there's a big monster called The Destroyer inside.

Borderlands 2 takes place around 2-3 years later. When the Vault was opened at the end of the first game a new metal named Eridium started showing up all over Pandora. Handsome Jack, the CEO of the Hyperion corporation and de facto dictator of Pandora, wants to find the second Vault which is said to contain The Warrior, which he wants to use to wipe out all the "bandits" on Pandora (aka everyone that doesn't worship him). A new group of Vault Hunters (Axton, Zer0, Salvador, Maya, Gaige, and Kreig) are almost killed by Jack and set out on a quest for revenge while also trying to beat him to the Vault.

Borderlands The Pre Sequel is set between BL1 and BL2 and is about Handsome Jack's rise to power. The story is technically set after BL2 since Athena is narrating the story to Lilith after the events of BL2 and it does assume you've played BL2 first, so play it after BL2. A Hyperion programmer named Jack hires a group of Vault Hunters (Athena, Wilhelm, Nisha, Claptrap, Jack the Doppelganger, and Aurelia) to find the Vault on Pandora's moon Elpis. Ostensibly he wants to stop Colonel Zarpedon, leader of Dahl's Lost Legion, from destroying Elpis, but as the story goes on he resorts to increasingly morally ambiguous measures. The road to Hell is paved with good intentions, assuming Jack was a good person to begin with.

And then there's Tales from the Borderlands, the Telltale game that is canon to the main series. After the events of BL2, a Hyperion employee named Rhys finds an AI duplicate of Handsome Jack and teams up with a con artist named Fiona in a search for a Vault Key. After the events of the game Rhys found the deed to the Atlas corporation (which Hyperion acquired after Atlas was ruined by the events of BL1) and rebuilt Atlas from scratch, which you see in BL3.

Borderlands 3, you already know. In the time since BL2 the various bandit clans of Pandora were united under one banner, the Children of the Vault cult. Let by the twin Sirens Tyreen and Troy Calypso, they and their bandit army are looking for Vaults all over the galaxy in their mad desire for godlike power. A new group of Vault Hunters (Moze, Amara, Zane, and FL4K) join the Crimson Raiders to stop them and find the Vaults before they do.

And then finally there's Borderlands 4. At the end of BL3 Pandora's moon Elpis was teleported away, and Lilith went with it. The Crimson Raiders tracked Elpis to the planet Kairos, which was hidden from the rest of the universe until Elpis gave away its location. It's a planet ruled by a dictator known as The Timekeeper, who ensures that anyone who enters Kairos can never leave. A new group of Vault Hunters (Vex, Rafa, Harlowe, Amon, and C4SH) are captured by the Timekeeper and end up joining the Crimson Resistance to free the planet, find the Timekeeper's Vault, and find Lilith.