I'd say that the first step is to work on composition/subjects. 

Professional have better materiel and skills, and there is also a lot of work editing the pictures, but if your subject is not interesting and your composition is boring the pictures will be bad no matter what. 

Look at pictures or paintings* that you like and try to understand how the different objects are placed - the strength lines they form, where they are in the picture...

When you take a photo of a foreground subject (like a person, an animal), try to have a nice background (and to wait patiently until any unwanted tourist has left the field !). Don't put your subject in the center, but try to put it at one third on the left or the right - unless you want to emphasis symmetries or stability. Have horizontal lines to give a feeling of stability, vertical/diagonal lines to give dynamism, and put your subject at the intersection of your main lines to direct the eye to it. 

Also an easy improvement is light : forget about pictures inside, they are almost always bad. Don't use a flash, artificial light is hard to use well. The sun is your best friend, but you should always try to have it in your back when you take a picture - that way the subject is well lighted and you avoid being against the light. Sunset light makes any crappy photo looks good.
Similarly, great color contrast helps a lot, but it is very hard to take a good picture when the subject and the background have similar colors.

*I'm not that much into still-life but I think they are great to look at compositions.

There are many factors but an understanding of composition and what camera settings affect lighting and thus the image: 

- Aperture: how wide the lens shutter opens to let light in. 
Wider = more light, higher exposure and more apparent distance between your subject and background, where your bg may even become blurry. Narrower = less light, lower exposure and less apparent distance between your subject and the background. 

- Shutter speed: how fast the lens shutter opens and closes. 
Faster = less light but can more successfully capture movement crisply. Slower = more light but can lead to more blurry photos depending on speed of subject.  

E.g if you're shooting a sports game during the day you can increase the shutter speed to avoid blurred images, knowing that you have sufficient light. If you're shooting at night, you'll want to reduce shutter speed to increase exposure, and opt to shoot slower subjects for a crisper image. 

-ISO: artificial exposure produced by your camera. 
This should be at the lowest setting and bumped only if you need more light to reach proper exposure. High ISO leads to noisier images

Better camera equipment allows you to more easily manipulate these settings (amongst others ) and produce higher resolution images. 

I was friends with a good though mostly hobbyist photographer (see here) for a while and we went photo hunting a few times. It was my experience that equipment helps to make more out of a photo but the key ingredients are

• an eye for photogenic objects and scenes
• experience how to capture that (e.g. from which direction/angle/time)
• general rules of composition and lighting

And only the last part you can fix with software later.

To learn the former, it probably helps to hang out with photographers.

I have picked up a few general tips from YouTube videos:

1. Significant objects are put on the intersection of the grid (3x3 or other)
2. You want the horizon to not look like it's the Inception movie, i.e it has to be horizontal.
3. It's better to make them darker than brighter. You can brighten the darker pictures, but the opposite is a worse operation.

Also, what I found from experience is that it pays off to find something to immobilize your phone/camera. That way, the details are not smudged. 

All of that is probably obvious or might be incorrect on professional inspection. A grain of salt is required here :)

For a more iterative approach that isn't guided by theory, you can do small experiments whenever you are taking a photo. When you are taking a picture of something, try any or all of the following and see which come out better: 

• Flash vs no flash.
• Move the camera up, down, left, and right. See 
• Move the camera closer or further away, possibly zooming to compensate.
• Move your subject to change the background.
• Try increasing or decreasing the amount of bokeh.
• If you have a friend nearby, try adding "off-camera flash" by having them hold up their phone flashlight. 

Over time, you can build an intuition for which of these things are likely to help.

My wife is a semi-professional photographer, and I can say that good equipment (including lighting) and post-editing are big parts of it in addition to just practice taking lots of photos. YouTube is probably the best cheap/low-effort option for learning this sort of thing.