You will have to talk with flight masters in most towns or cities you visit in order to gain access to that settlement's flight path. Flight masters who have a stop you have not yet unlocked will have a green displayed above their head (and on the mini map), much like a NPC with a quest. Unlocking flight paths as you progress through various zones will allow you to pay for a quick ride to a destination you have previously visited. This method of travel is significantly faster than walking or riding a mount.
You will see a resurgence of power leveling services, item trade services, and gold farming. Classic will have a fresh economy meaning for your average gold farmer time spent in classic is way more efficient. It will probably be worse than it was back in the day. Just look at the blizzard group finder on retail, littered with advertisements for m+ boosting websites. Disgusting. There will be so many gold farmers blizz probably won't be able to keep up. And this is what I truly fear because if the gold farmers start to win activision will step in and introduce wow token. And classic will die.
The quickest way to level up is to travel through the game's zones, completing all the quests you encounter along the way. Completing quests grants you Experience points, which will slowly fill in your experience bar with blue or purple. When the bar fills, you reach the next level. Alternatively, you could choose to run Dungeons or Battlegrounds, which are not covered in this beginner's guide.