How to set Nikon d5600 to bust mode?

I just got the camera and I don't know what button to hit.

READ the MANUAL.

In the back of the manual is a section called "Index". Look up "Burst" (not "bust") and that will refer you to the page number where it will discuss the burst mode. It is also called "Continuous (Release mode)". There's more information in the manual on burst mode than we could write here.
If you lost the manual, download a PDF version from the Nikon web site.

RTFM!

Just hold down the shutter button. It will start shooting continuously. You can adjust the rate of the continuous shooting through the menus.

Link to the manual:
http://download.nikonimglib.com/archive2/J3SEU00Dlpa302wk5qq71fVTHF73/D5600RM_(En)01.pdf
Page v Table of Contents says Continuous shooting (Burst mode) is on page 76.
The Index (page 391) says Burst is on page 76 and Continuous (Release mode) is on 75 and 76…

If you can't get your mommy or daddy to read you the instruction manual, just use a hammer or a baseball bat.

Congratulations, you got a VERY NICE camera, and once you learn to operate it you will have lots of fun experiences with it. The first thing you must do is read the camera's Instructions Manual; very, very important. Have your camera next to you so you can refer to it as you read the Instructions Manual.
IF you don't have an Instructions Manual, get it free from the Internet on PDF, download it to your computer and make a hard copy for you to carry around. READ it so you can learn and discover the many, many features, functions and options the camera has to offer you. It will explain what the features are and how and when to use them to get maximum use of this great camera. You can read a few pages every night before going off to sleep or while you're sitting in front of your TV, and read during commercial advertisements (do you really pay attention to commercial ads on TV?)
Look at your camera as if you were going to shoot a photo. On the top left corner, you will see the knob with small letters on it (I have have a different Nikon camera model with similar features); the letters should read something like, M A S P (Manual Mode, Aperture Priority, Shutter Priority, Program) and just beneath that is a flat wheel that also has letters on it that start with S (for single), Cl Ch… Etc. Now look at your Instructions Manual's INDEX in the back and look for Burst Mode and also look up Continuous Release Mode.
Why do you want to machine gun your camera? You think it's "cool"? It really isn't to more experienced photographers. And you'll have a boatload of images to sort through afterward… Do you want to waste time going over a bunch of photos that look the same, or do you want to develop the skill to get the right shot when it counts? Think about that. I'm "Old School" from the film era, I don't machine gun my camera unless I'm photographing particular sports, like Gymnastics, or Track & Field and it's usually three shots and rarely 4 IF there's enough light to do so. You're just beginning; get into GOOD habits, not because they're "cool" but because they serve a purpose. Please READ YOUR INSTRUCTIONS MANUAL.
The VERY BEST advise and suggestion I have after "Read your Instructions Manual," is: take a formal organized Beginner's Class. You will enjoy the class (it's a fun class) and you will save yourself many, many months of reading and frustrating and often discouraging trial and error experiences; that's why we say it's a shortcut to learning.

Handbook? And indeed "burst" mode, unless you are intent on photographing women's chests!

It's BURST mode.
READ YOUR FREAKING GOD DAMNED MANUAL.
Lazy but.

"Bust" mode. Eh?
Does that mean a head - and - shoulders shot,
or damaging the camera?
I guess if someone was ugly enough, it would be possible to achieve both at the same time.
Anyway,
Read
The
Owner's
Manual.

Wow. Such incredibly nasty comments, however, purchasing a camera, that actually comes with "NO" manual is an issue. When you are a "first time" camera buyer, and trying to "LEARN" to actually get their camera to do what they want it to. Asking questions is usually a good way to find out how things work, and learn from it. HOWEVER, judging by some of the comments on here, I believe I will pass. I have "no desire" to be ridiculed for my "lack of knowledge".

Agree with you - A frightening bunch of small-dicked people responded. Like you, I can't see how to do a burst shoot. Shite documentation from Nikon, absolutely appalling behaviour from fellow users. Stuff you all!