In almost every single Taekwondo school forms are practiced. They are an essential part of the taekwondo syllabus. However, in some schools they may be more or less emphasized and the purpose of training is slightly different. Here are a few examples.
In a Taekwondo school or club that is heavily focused on competitive sparring (like Olympic Style or WTF style sparring) a student may rarely or even never practice forms. Maybe they are a cursory add on for belt testing. Although, a competitive school also practice forms a lot if they are entering into Poomsae competitions. For example, when I was coaching at the University of Pennsylvania we trained for sparring a lot and we trained forms a lot since we competed in both of those areas.
A more formalized school may practice forms a lot and for a variety of reasons: testing, tradition, workout, technique application, etc.
A school focused on practical techniques may practice forms a lot or very little, it depends on how they interpret forms.
So, before you endeavor to improve the quality of your forms have the big picture and your end goal in mind. Steps to improve your poomsae competition score may not be the the same as improving your hyung for belt testing.
#1 -- Memorize the form. Learn it from your instructor or from higher belts. Watch them do it, watch videos, read books, and repetition. If you don't have the form memorized, you will be "thinking" instead of doing.
#2 -- Practice Visualization. Once you have the form memorized, close your eyes and picture yourself executing all of the moves.
#3 -- Work on your steps and stances. Hold your belt, or place your hands behind your back, then move through the steps of the form in sequence. Try to step as fast as you can and land in a perfect stances every time
#4 -- Work on your blocks and strikes independently. The taekwondo forms consist of not too many blocks and strikes. Drill those techniques on their own. You can drill them from riding horse stance, front stance or back stance. You can drill them in a stationary way in front of a mirror or you can drill them while stepping forward and/or backward.
#5 -- Realize that two things make a Taekwondo form look good. Speed and correct positioning. How fast can you execute the technique? When you complete the technique, did you land in the correct position? Are your feet/stance correct? Are your arms in the correct position? Your hips and torso? You head, shoulders and gaze?
#6 -- Realize that "power" comes from the momentum of your entire body, not just speed. If you step then punch or block as fast as you can, that is not enough. You must step AND punch or block as fast as you can at the same time.
#7 -- Start developing "intent" with your poomsae/hyung practice. What is intent? Intent in martial arts practice can be described in many ways. Here are a couple: 1. It is your will while practicing. 2. It is knowledge of your enemy 3. It is understanding of the application of technique during execution. I think all three are valid and work together.
#8 -- Make it more difficult to practice. There are several ways to do this. Remember, the mind and body responds to the need for greater exertion. Here are some ideas: 1. close your eyes. 2. Practice in an unfamiliar place 3. Practice on an uneven surface (like sand, rocks, or a hillside). 4. Practice in hot or cold conditions 5. Practice with someone else trying to hit or shove your. 6. Practice with a weighted vest.
These are just some tips. Poomsae practice and perfection is a lifetime endeavor. Find a good instructor and commit to the life time journey that martial arts training is.