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First off, sprites from The Spriter's Resource since half of Bulbapedia's are animated.
To get started, open Paint. Yes, MS Paint. You'll see.
Shove your sprites in however you want.
RIGHT CLICK using the Eyedropper tool to select colors. This will save that color as the secondary color in-use. The eraser should fill in with that right-click color.
I highly recommend constructing the palettes first. This will make things really easy later. I use the eraser on the smallest size.
Once you have palettes set up, the rest is easy. Using the Eyedropper, LEFT CLICK on the color you want to be REPLACING, grab the eyedropper again and RIGHT CLICK on the color that you want to REPLACE IT WITH.
To repeat, put the color you want gone in the primary color spot, and the color you want it to end up in the secondary.
Now RIGHT CLICK over the sprite with the Eraser tool. This will replace all color of the primary with the secondary color.
When constructing your palette, I recommend replacing all of the old color with some filler color so you don't miss anything.
For example, the third color on Spoink's palette only appears on TWO PIXELS on the entire sprite. You don't want to miss that shit.
You may want to copypaste every single sprite's palette underneath it. This lets you select things quicker. You can replace the colors in the palette as you replace the sprite's colors, but make sure you get everything in the sprite, as you will no longer have anything to compare missed colors to.