I have been shooting for six months now using my Nikon D5600 and the 18-55 mm kit lens.

You would probably benefit from using a prime for a while. Pick up a 35mm or 50mm prime. Then shoot with it as much as you can for the next 6 months.

• It's been claimed that working with a prime for an extended time helps develop composition skills.

• With some zooms, by the time the lens is stopped down enough to reduce aberrations, diffraction has already set in.

• It's easier to work in the sweet spot of fast primes without sacrificing on other settings than it is when working with zooms.

What can I do to improve the quality? ... mine could not be distinguished from a high-end smartphone.

• Select settings conscientiously.

  • Your first image was shot at F10. The high F-number (smaller aperture) produces deeper depth of field at the expense of more visible diffraction. The DOF makes more of the image look in focus, but the diffraction sacrifices fine detail. For crop sensor, F5.6 is a good place to start. Adjust up or down as desired.

  • Try to "nail" exposure in camera. Underexposing and attempting to recover in post sacrifices "quality". Images processed this way may look like they were taken on cell phones because it mimics the exposure problems of small sensors.

    Use the histogram to evaluate exposure. Typically, it should be spread out over the entire range. You might want it to lean to the right or left, but crashing into the sides represents lost data.

    The following histogram is taken from one of your images. Crashing into the left represents lost shadow detail. The empty space on the right indicates poor use of available dynamic range.

    

• Perception of quality can also be influenced by brightness, contrast, and color. Consider this image...

  

  • The first third is too dark. The shadows are completely blacked out, and there are no highlights.

  • The middle third is "brightened" by adjusting curves. Similar effects can be achieved by increasing exposure or gamma. This makes better use of the available dynamic range and increases the detail that can be seen in the image.

  • The final third has additional curve adjustments to increase blue and decrease red. This makes it look "cleaner". The sky doesn't look as muddy. The foliage is greener.

I use Photoshop to do the color manipulation for my images.

Try to get as much done in camera as you can. Your images will be better, and they will be easier to edit, if it's even necessary. Shoot RAW+JPG, but focus on getting good JPGs out of the camera. Reserve editing for final touch ups. Resist the temptation to think, "I'll just fix it in post."

Even if your intent is to ultimately process RAW, you should still aim for good JPGs. In the field, the JPG is a preview of the quality of the data you're capturing. Better JPGs mean better RAWs. Without a good base, results from RAW processing will still be suboptimal.

When I finish I export them in PNG format and upload them to Google Photos, then I upload them to Instagram.

I suspect you're uploading PNG files because you heard somewhere that they're "better". For local storage, they're fine. Go ahead and save in TIF, PNG, PSD... But...

• Everyone runs low on disk space eventually. Consider lossless HEIC for local storage of images that you are finished editing.

• PNG gives you more of what you have. It doesn't make anything "better". The photos you wish to emulate are mostly presented in lossy formats, like JPG.

• Google recompresses photos to JPG on the free tier, so uploading PNG just wastes bandwidth and upload time for zero quality gain.

• If you pay for cloud storage, you'll spend more than double with PNG than you would with lossless HEIC.

• If you're resizing images to get PNG file sizes down, you're losing more "quality" than you would by saving directly to JPG or HEIC.

Here are some file sizes for comparison:

• Camera JPEG = 10.9MB
• JPEG minimizer = 0.8MB
• HEIC-50 = 0.9MB (iPhone appears to use a setting in the 50-55 range)
• HEIC-Lossless = 11.8MB
• PNG = 27.6MB

Upload JPG (or HEIC) files with appropriately chosen compression settings. A collection that would take an hour to upload in PNG, takes less than 2 min with minimized JPG or HEIC. Lossless HEIC would take less than 26 min.

See also:

• Why do people recommend 50mm or other prime lenses as starting lenses for learning photography?
• How and why do you use an image histogram?
• What is the "optimal" file size of JPEG images with respect to their dimensions?