Just by changing how you hold your smartphone you can improve your photos dramatically.
Hold the camera with both hands
You are not as steady as you think. When you hold the phone with one hand it will shake. When you zoom in on one of your old photos you will see that it isn't as sharp as it could have been.
Hold the camera close to your body
The farther away from your body you hold the phone, the more your arms will shake. Keep your upper arms to the sides of your body. Hold the phone close to your eyes so you see the details of what you are going to shoot.
Hold the camera horisontally
You do not want the horizon to cross the photo diagonally. Align it with the upper or lower edge of the display. Many camera apps have levels to help you.
Shoot parallel to the ground
This strategy has many advantages. You are forced to get to the same level as whatever you are shooting. Kids are short. Get down to their level to get their perspective. Your phone lens should be on the same level as the middle of what you want to be in the middle of your photo. When you take a headshot the camera lens should be in level with the eyes. If you take a full body portrait the lens should be at about waist level. When you shoot a toddler crawling on the floor, you too should be crawling on the floor. When you shoot a building that you want to capture whole, step back until you get it all in the frame still shooting parallel to the ground.
Use the on-screen shutter button, not the volume button
Yes, it was presented as a big improvement when you could use the volume button as a shutter button. In reality it was a step backwards. You have to use more force to trigger the shutter with a physical button. The on-screen shutter button only needs a slight touch. Guess which one are more likely to make you shake at the critical moment?
If you buy a phone case with a lens mount, buy one without a hardware shutter. You will not be using it anyway since it is more likely to ruin your photos. Moment photo case is a great choice.
Lean against something
It is a bad idea to stand on one leg while you shoot. Two legs are better and three is best. That's why there are so few bipods and so many tripods. The less chance of shaking, the better. Lean against walls, fences or whatever you have available. Squat if there is nothing to lean on. Or use a proper tripod. If you get a Shoulderpod G1 you can mount your phone on any standard tripod. It's also a great grip for holding your phone without risking a drop.
Break all the rules
When you know the rules and understand why you use them, you can experiment with breaking them. By following the rules you will take better photos. By breaking the rules intentionally you can take master shots.