It's my first time leveling on The Lost Isles and a guide I'm following suggests to go to Azshara next. How do I get off of the Lost Isles and get to Azshara?
You can naturally leave the lost isles at around level 12 when the quest chain completes. This shouldn't be very long. Goblins, like Worgen and the class Death Knight are locked in a story (phased) zone for much of their early levels.
If you want to meet up with friends early, you need to play a different race to start.
It is possible after a certain point in the story to be summoned out of your zone with the help of a meeting stone or warlock if you level too quickly, however you will forfeit any remaining quests in the area as you can not return. I wouldn't recommend this action as the quests here are more exciting than the rest of the world - they are even voice acted!
The only way to get off the Lost Isles is to complete the quest line. Any attempt to leave either through summoning, join battle ground or joining a dungeon will result in a message saying you need to complete the Goblin starting quest line. Once finishing the goblin quest line you will find your self at Orgrimmar and can go where ever you want as normal.
I discovered a way by accident. I was just questing i was about a level 6 goblin. And my laptop over heated so i had to exit the game completely, then restart to game and when i restarted the game i used my hearthstone to take me back to the questing area, but it instead took me to orgrimmar. Hope this helps!