The dog looks like it's floating most of the time. I can't tell if the animation is meant to be a run or walk or something else, because the poses don't look particularly like any of them. It looks like you're reusing the jump poses? I recommend making some unique poses for each animation, even though it's more work. If you look up "dog run cycle" online, you should get lots of results, including some photos like these:
one,
two. Your animation doesn't need to have these many frames, you can get away with 3-4, as long as they include the extremes of the pose - the leap with both legs off the ground and apart, and when both legs are together under the body. There are similar resources for walking animations, if that's what you're aiming for.
Since you have a dark outline on the dog's paws, the dog always appears like it's floating 1px off the ground, which doesn't help with the floaty look. For the animation frames where the dog is on the ground, have the legs go all the way to the ground, without an outline in the middle. It'll help it look much more grounded. The sprites might look weird in isolation without the outline, but you have to consider the final look. No one's going to be looking at the sprites in isolation.
The environment art looks nice, but rather bland and flat. You've got castle walls and some brown dirt with grass on it, these appear in just about every generic tileset and tell me nothing about your game's world, or even genre or general mood. Even if a generic setting is the goal, you can make it look unique and convey a mood through
how you draw them. You've drawn a very generic brick pattern, for example, but the bricks look neither old and cracked nor new. You have crystals and some tiles for vines growing on the bricks, but nothing about the other tiles suggests that the place is abandoned or what might've happened to it.
The light brick surface on which you walk looks nice, but has the same problem of telling us nothing. At a closer look, the brick also seems to be melting? That might be cool, if it's intentional. If it's intentional, make it more obvious perhaps, with larger gloops of stone? And if it's not melting, then what is happening to it? I thought at first it was just cracked and rough (unlike all the other bricks xP), but then I noticed some tiles have the brick dripping down onto lower bricks.
In short: You seem to be going for an overgrown fantasy castle look. Give some thought to how and when it became like this, and integrate indications of this into the other tiles, even those that don't directly show plants. If it's been a while and it naturally got like this, make the bricks less perfectly even, add some cracks. If it magically suddenly became like this, make the bricks tidier and brighter to show their newness.
Speaking of integrating the plants and bricks: try having the moss, vines, and grass follow the brick lines a bit more. Moss and grass especially tend to favour upward-facing surfaces, so they'll often have grown further in the bits where the mortar is gone, and just on the upper edges of the bricks in general. This can make help them look like they're actually growing on the bricks rather than just merely slapped on.
The ground tiles have very noisy edges, especially compared to the clean geometric pattern within. The edges should feel like they're consistent with the front, they should have the same scale of detail. The ground overall feels out of place with the bricks and the detailed grass on top of it. The bricks seem to be going for an only mildly stylised texture and get dark towards the middle, while the ground has a Sonic-like highly stylised pattern, and gets dark at the bottom. They don't look like they belong in the same game. Then you also have the background ground/stone(?) texture, which is just a pattern throughout without going solid anywhere. Notice how Sonic, when it does use its patterned textures (mostly in Emerald Hill Zone), uses them throughout the map, and maps using more realistic textures have such textures throughout (and the more "realistic" textures are still very geometric, which maintains a sense of consistency).
I think the stylisation on the ground looks cool, but if you're going for that kind of look, commit to it, stylise everything else too. And if you want a less stylised look overall, then don't stylise the ground either, unless you have some story reason for it that's explained to the player.
In your maps, it looks like you have lighter tiles behind the dirt, so the dirt looks blockier than it has to.
Lastly, watch out for the tangent of the lowest crystal on the second tree in the tileset. The way it's placed exactly 1px into the trunk all around makes it look like it has a dark outline.
Edit: I completely forgot to talk about colours! If you don't have a specific look in mind and don't feel confident in choosing colours, then starting with a palette like that is good, since it helps you get started. However, don't feel like you have to use only those colours, feel free to change or add colours if you think it helps your specific art look better. You might end up with something very different at the end, or with largely the same palette, and the only thing that matters is that the resulting art works well.