Super Mario Bros custom tileset

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I tried to illustrate a few points.



Your colour choices could be better. Most, if not all of your colours have a saturation of 240 - you can use lower values, too.
Also, hue shifting is a good technique to make your shades look better. For example, darker shades of red and blue can be slightly more purple, darker yellow shades more orange or green, darker greens more blue.
Furthermore, you should avoid using multiple colours that look almost the same. For Goomba, it would be the two shades of grey used for the eyes and the bright brown used for the belly. Generally, make sure your colours have good contrast. If a colour looks like another one, you do not need it. Your shading should be visible.

Then there's the shading issue - most of these have no defined lightsource, you just follow the outline with your shading colour. This usually makes sprites look bad instead of giving them volume and depth.

Basically, take a look at the spriting dictionary that has been linked earlier.