This is a tutorial on how to make custom palettes, and good ones at that.
First, you want to click on "Level" in the toolbar.
As of Lunar Magic 1.7 or 1.8, you can actually just click the blue star on the LM toolbar inbetween the Overworld Castle and the pink Berry.

Then, you want to make sure "Enable Custom Palette"(The only option in the first section) is checked.
Now you need to click on "Change Palettes and Edit Colors"(First option in the fourth section)
This is your tool to make custom palettes.
It should look something like this.

Those boxes should only be editable when "Enable Custom Palettes" is not checked. If you want different base colours, make sure "Enable Custom Palettes" is not checked, change the boxes to your liking, then check off "Enable Custom Palettes" again.
*Note: Mario and Luigi's palettes are only editable when the "Enable Custom Palettes" is not checked, except for the three gold colours, black and white.
LIST OF PALETTES AND CONTENTS
Now that you have this fancy little window open, you should know what each palette is and what they are used by.
First thing you should know is every horizontal line is one palette, and within those palettes are the colors, which each have their own hex number/letter.
Here is a list of palettes, and what is in each palette.
Object Palettes;
Palette 0; A background, status bar
Palette 1; A background, status bar
Palette 2; Default main Foreground, red berries
Palette 3; Various objects(first few colors change), pink berries
Palette 4; Gray/light blue used for cement block and others, green berry
Palette 5; Green, uses for pipes and others
Palette 6; Gold/Brown, used for most "blocks" that Mario can hit, pipes
Palette 7; Red and Blue palette, used for Pipes, Nets, Water, Red and Blue Switch Palace switches and their corresponding blocks.
Sprite Palettes;
Palette 8; Mario and Luigi's Palette.
Key, Keyhole, Only the three golds, the second black, and the white colors may be changed when using Custom Palettes.
Palette 9; Silver/Gray used for Bullet Bills, Boos, Red Berry sprite,
Palette A; Orange/Yellow, used for Goombas and others, Feather powerup, pink berry sprite, same use as red berry sprite
Palette B; Blue, used for blue koopas and some others, green berry sprite, same use as red berry sprite
Palette C; Red, used for red koopas and the Mushroom powerup and some others.
Palette D; Green, used for green koopas Fire Flower powerup, Chargin' Chucks
Palette E; Diggin' Chucks' rocks
Palette F; Used by Dino Rhinos and Dino Torches, and uneditable but used by Magikoopa and Big Boo
ACTUAL PALETTE EDITING
'kay, this is where we begin actually changing the palettes.
Basically, you just click on a color, and change the settings.
I most often just change the spot of the color, like moving this

to over here, in the green area

Now that green doesn't look that pleasant, but it isn't eyebleeding.
Think of it as an "ok" green.
People have different opinions of what is "ok" and what is "good", but try to get something that looks nice and ask people's opinions of it. If just a person or a few don't like it, but some others do, keep the color as it is if you like, as it is very rare for everyone to like the same thing.
Now that you have changed the color a bit, you can change the darkness and light of it. To do that, just click and move your mouse over the bar on the right, until you get a nice shade.

You do not want palettes that look like this.
You want a nice looking palette, like the one below.

See how nice this palette is compared to that other, barfy one?
HELPFUL THINGS
Easy Gradient; Hold Alt key, then right click the leftmost colour, then click the right most color.
Creates a gradient with the colors inbetween.
You want to make sure the left color is darkest, and right color is lightest(or the other way around, if the graphics are set up that way)
If you don't it will most often look ugly.
Example;

First few colors of Palettes 0 to 4.
Replacing Colors; Left click on a color, then right click on the space of another color.
The first color in ALL the palettes is uneditable since it is the transparent colour.
Colors 4 in palette 6, and D of palette 6 and 7 on the overworld are uneditable, as they are the animated, flashing colours used by things such as Yoshi Coins and the level icons on the overworld.
You can apply a hex edit to disable the flashing so you would be able to use them like normal colors.
You usually want your background to be darker than the foreground, as it's not as close to you as the foreground is.
You also don't want the background using the same or almost the same color as something in the foreground and vice versa.
It's usually better if your graphics are shaded.
Most of the time unshaded graphics are ugly and not detailed.
Anything to add or questions will be appreciated.
First, you want to click on "Level" in the toolbar.
As of Lunar Magic 1.7 or 1.8, you can actually just click the blue star on the LM toolbar inbetween the Overworld Castle and the pink Berry.

Then, you want to make sure "Enable Custom Palette"(The only option in the first section) is checked.
Now you need to click on "Change Palettes and Edit Colors"(First option in the fourth section)
This is your tool to make custom palettes.
It should look something like this.

Those boxes should only be editable when "Enable Custom Palettes" is not checked. If you want different base colours, make sure "Enable Custom Palettes" is not checked, change the boxes to your liking, then check off "Enable Custom Palettes" again.
*Note: Mario and Luigi's palettes are only editable when the "Enable Custom Palettes" is not checked, except for the three gold colours, black and white.
LIST OF PALETTES AND CONTENTS
Now that you have this fancy little window open, you should know what each palette is and what they are used by.
First thing you should know is every horizontal line is one palette, and within those palettes are the colors, which each have their own hex number/letter.
Here is a list of palettes, and what is in each palette.
Object Palettes;
Palette 0; A background, status bar
Palette 1; A background, status bar
Palette 2; Default main Foreground, red berries
Palette 3; Various objects(first few colors change), pink berries
Palette 4; Gray/light blue used for cement block and others, green berry
Palette 5; Green, uses for pipes and others
Palette 6; Gold/Brown, used for most "blocks" that Mario can hit, pipes
Palette 7; Red and Blue palette, used for Pipes, Nets, Water, Red and Blue Switch Palace switches and their corresponding blocks.
Sprite Palettes;
Palette 8; Mario and Luigi's Palette.
Key, Keyhole, Only the three golds, the second black, and the white colors may be changed when using Custom Palettes.
Palette 9; Silver/Gray used for Bullet Bills, Boos, Red Berry sprite,
Palette A; Orange/Yellow, used for Goombas and others, Feather powerup, pink berry sprite, same use as red berry sprite
Palette B; Blue, used for blue koopas and some others, green berry sprite, same use as red berry sprite
Palette C; Red, used for red koopas and the Mushroom powerup and some others.
Palette D; Green, used for green koopas Fire Flower powerup, Chargin' Chucks
Palette E; Diggin' Chucks' rocks
Palette F; Used by Dino Rhinos and Dino Torches, and uneditable but used by Magikoopa and Big Boo
ACTUAL PALETTE EDITING
'kay, this is where we begin actually changing the palettes.
Basically, you just click on a color, and change the settings.
I most often just change the spot of the color, like moving this

to over here, in the green area

Now that green doesn't look that pleasant, but it isn't eyebleeding.
Think of it as an "ok" green.
People have different opinions of what is "ok" and what is "good", but try to get something that looks nice and ask people's opinions of it. If just a person or a few don't like it, but some others do, keep the color as it is if you like, as it is very rare for everyone to like the same thing.
Now that you have changed the color a bit, you can change the darkness and light of it. To do that, just click and move your mouse over the bar on the right, until you get a nice shade.

You do not want palettes that look like this.
You want a nice looking palette, like the one below.

See how nice this palette is compared to that other, barfy one?
HELPFUL THINGS
Easy Gradient; Hold Alt key, then right click the leftmost colour, then click the right most color.
Creates a gradient with the colors inbetween.
You want to make sure the left color is darkest, and right color is lightest(or the other way around, if the graphics are set up that way)
If you don't it will most often look ugly.
Example;

First few colors of Palettes 0 to 4.
Replacing Colors; Left click on a color, then right click on the space of another color.
The first color in ALL the palettes is uneditable since it is the transparent colour.
Colors 4 in palette 6, and D of palette 6 and 7 on the overworld are uneditable, as they are the animated, flashing colours used by things such as Yoshi Coins and the level icons on the overworld.
You can apply a hex edit to disable the flashing so you would be able to use them like normal colors.
Code
02629 $00:A429 12 bytes Palette Changing [B9 0C B6 8D 22 21 B9 0D B6 8D 22 21] to [EA EA EA EA EA EA EA EA EA EA EA EA] will disable the flash animation of color #64. Make sure that the "View Animation" function in Lunar Magic is disabled. WARNING: Will disable the animated colors on the OW too!
You usually want your background to be darker than the foreground, as it's not as close to you as the foreground is.
You also don't want the background using the same or almost the same color as something in the foreground and vice versa.
It's usually better if your graphics are shaded.
Most of the time unshaded graphics are ugly and not detailed.
Anything to add or questions will be appreciated.