Online games are not much different than home play. There are a couple of options. One is where you run through the published adventures. Currently Lost Mine of Phandelver, Hoard of the Dragon Queen, Rise of Tiamat, or Princes of the Apocalypse, and Curse of Strahd. If you’d like to run the D&D Adventurers League modules, your DM should go to the Dungeon Masters Guild and download the needed adventures. In either case as long as the Dungeon Master runs the adventures according to the D&D Adventurers League Player’s Guide and as long as the players record their rewards on logsheets their characters are transferable to any D&D Adventurers League game, anywhere in the world.
If you’re playing online we encourage you to live-stream. Live streaming is using free-to-watch streaming service, such as Google Hangouts On Air or Twitch.tv or similar apps.
To play online you can use any of several popular virtual table tops, such as Fantasy Grounds (officially licensed by Dungeons & Dragons), Roll20, D20Pro, or any other online app.