The question of fast travel is likely to be at the forefront of your mind, right from the start of Return to Moria. You'll find a Ruined Mapstone at the first camp but it needs repairing with a Black Diamond. As you won't have access to these until a little later on, you'll need to rely on your legs—and pickaxe—to get to where you want to go at first.
When you do unlock fast travel, you're still going to need to do a lot of running around but you'll no longer need to backtrack all the way into previous areas if you've left something behind. With that in mind, here's how to unlock fast travel in Return to Moria.
Return to Moria fast travel: How to unlock
You need at least two working Mapstones to fast travel between their locations in Return to Moria. The first of these—a broken one—is found in the initial camp during the tutorial section. You might have ignored it because it requires a Black Diamond to repair and you won't have access to these until you reach the Elven Quarter at the earliest. Even if you do somehow manage to find a diamond and repair it this early in the game, you'll need a second active Mapstone to travel to anyway.
While there are likely other ways to get them, the only two methods for acquiring Black Diamonds I've encountered so far are either looting Orkish Chests from the Mines of Moria region onwards, or by having one drop during a horde attack. You'll encounter your first horde when you activate the Forge of Narvi in the Elven Quarter.
The second Mapstone is available once you reach the Mines of Moria, though this one also needs to be repaired. As each world is procedurally generated, I can't direct you to it but you should find a second ruined camp in the area, similar to the first one.
You can also build your own Mapstone at any camp you've restored, but it will cost three Black Diamonds as well as 20 stone. Be careful about wasting your diamonds if you don't have any stockpiled. You never know when you're going to stumble on another Ruined Mapstone in need of one.