In WoW Classic, leveling is a bit more difficult. You don't expect to level multiple times in each area and then move on, and some areas are better for leveling than others.
Either way, each race is stuck in its start area, so you have no choice. It's not that big of a deal for humans and night elves, but dwarves and gnomes should leave Kharanos as soon as they can.
World of Warcraft Classic: Best Alliance Upgrade Zones
Once out of your starting area and the area immediately outside, you have a few options. In order, Darkshore (11-19), Loch Modan (10-18), Westfall (9-18) and Duskwood (10-30) are all solid starting points.
Keep these ranges of levels in mind. Duskwood is a great area, but going straight there at level 10 can cause problems with wasted time dying on high level monsters that you accidentally venture into.
Westfall is a key area because it contains the Deadmines, one of the first dungeons you can access. Quests are also relatively lenient in the area, making this an easy area to clean up.
Loch Modan is a fast and furious area full of effective "kill these" quests that you can complete multiple at a time with very dense spawns. As long as you avoid getting harassed, it will be easy to get your level up and out of it. Be careful not to wander the Badlands (36-45) in the south and get killed by high level enemies or Horde players.
Darkshore is roughly the same, if not slightly higher in average level. The quests are even thicker on the floor here, making it the best area for this range of levels.
Once you've explored the contents of most or all of these areas, you should be ready to take on Redridge (15-25), which should get you to the point where you can complete the higher level Darkshore quests as well. .
At this point, wetlands (20-30) are pretty much the only place you can go. There aren't a ton of quests here, and most are clustered near the east of the map. This area is meant to be a final Alliance leveling area before you have to venture into a contested area ... which is a bit of a different ball game. Just make sure the one you're venturing into isn't the Burning Steppes (50-59) to the north; at 30, no luck on your part.
Keep these areas in mind and you will go through low levels: and be ready to take on the contested areas.
