Head on over to Documents and download the AL Dungeon Masters Guide (this is not the same thing as the D&D DMG hardcover book). It includes information needed by Dungeon Masters in order to run AL-legal games.
Most Adventurers League adventures are one-shots. These are short adventures designed to be completed in 2 or 4 hours (usually). AL-legal adventures can be purchased and downloaded from Dungeon Masters Guild.
Important: Not everything on Dungeon Masters Guild is AL-legal. Make sure it has the Adventurers League logo on it.
If you are looking for some adventures to run as a first-time AL DM, these have been recommended in the past.
DDAL05-02 The Black Road (5e) - Wizards of the Coast | Dungeon Masters Guild
The Last Resort WBW-DC-TLRMB-02 - Dungeon Masters Guild | Dungeon Masters Guild
Sisterhood of the Blade (CCC-UNITE-AN-04) - Dungeon Masters Guild | Dungeon Masters Guild
Most official WotC D&D hardcovers have been made AL-legal. Download the Adaptation Guide from Documents to see the list of available hardcovers and the modifications (if any) required to make them AL-legal.
Important: If a hardcover is not in the Adaptation Guide, then it’s not AL-legal.
AL does not allow homebrew or custom adventures. You cannot invent your own settings, classes, races, magic items, etc. Everything must come from allowed sources per the AL Players Guide and AL Dungeon Masters guide.
If you want to create your own adventure, you can create a DC (Dungeoncraft) following the DC guidelines found in Documents.
Adventurers League games can be played anywhere. Run your game at your FLGS, on Discord, at your school, or in your home. As long as players bring AL-legal characters and you play by the rules of AL, then your game is AL-legal.