ctrl + wasd + space.

hold w to forward, place pinky on ctrl to crouch mid movement, while ctrl is pressed use tumb to tap space to jump forward, release ctrl while in jump, repeat to jump through the whole map.

Edit: for all replies mentioning rolling and sprinting in between for extra speed, you are right, but one comment explaining that is enough.
for all replies saying that rolling on the floor is faster then bullet jumping, you like Ivara, I get it, shes cool, stop.

Bullet jump everywhere, use aim glide to adjust so you don't hit ledges. It just takes time to get used to the movement. Plus a lot of folks probably know the tilesets well enough to be able to tell where they need to go to get to the next tile.

TL:DR Basically it's a combination of bullet jumping, rolling, and gliding.

In case you didn't know how to bullet jump, hold down the crouch button (ctrl if you're on PC), look at the direction you want to go and press jump (space bar for PC). You can use this ability to access hard to reach places that are vertically above you as well.

While you're in the air during your bullet jump, performing a roll will make you go even faster. And gliding just helps with crossing large valleys horizontally so you don't fall to your doom. For tight spaces, it's mostly rolling around like those destroyers from Star Wars.

Jumping while crouched will let you Bullet Jump, which is how people are getting so quick.

You can chain bullet jumps and slides together (slide, bullet jump, crouch before you hit the ground, hit the ground sliding, bullet jump again).

Once you get more into it, you can start rolling in midair with a directional key held to give yourself magical ninja momentum in that direction, even if it isn't the direction you're currently travelling.

W + slide + jump jump + roll

I use Bullet Jump + Roll + Aim Glide. I also use Mobilize frequently.

Combo things mid-air. Start with sprint. During sprinting hold ctrl for slide. During sliding press space to bullet-jump. You can bullet-jump only once per actual ,,liftoff" off of the ground and it resets on contact with any surface (floor, walls) This will be your main and easiest method of quickly moving around. Now for something better. If you're already in the air you can press the zoom button to aim glide. This will give your jumps quite a large range and speed. If you see you're starting to slow down during the gliding part just press roll button without holding any movement keys. It will give you that last little push you sometimes need. If you happen to press any movement buttons while rolling out of the aim-glide you will do a little flip and change direction. Yes, you can even roll backwards while flying forward. Kind of a handbrake.