Trust me, I've been there. I'm a designer/programmer first and an artist far last (pixel art is the only art I'm decent at.)
First of all, your main character is kind of strange looking, and doesn't really look as much like a mouse as a small elephant (unless that's what they're supposed to be, then I would make the trunk curve downward). Round out the nose, make the ears smaller. Make the jumping pose more active (his legs shouldn't be together when he's jumping. See: http://gamerfitnation.com/wp-content/uploads/2011/08/mario-jump.jpg
For the tiles, try to give them more texture. Rock patterns are probably the easiest pixel art thing to do. Outline a bunch of shapes with your darkest ground color, then put your darkest ground color along the edge of each rock, then your lightest near the top. Similar applies to grass tiles, but you need to be more careful with the shape. The point is textures are made up of discrete objects (rocks, grass, ect...) which each have mid-tones, highlights and shadows.
That cheese is clearly a re-sized sprite. Redraw it to fit with the scale of everything else so that it doesn't have that blocky look.
That bottom screenshot is by far the worst looking. Bright colors, no shading, sloppy gradients, weird particle effects (I assume that's what they are).
I'd recommend giving those tutorials a try. They're pretty easy to follow, and will help you improve the polish of your game.