I have heard that muscle soreness happens after 24-72 hours after exercise, however most of the time after exercise immediately my muscles get sore, is this considered as muscle soreness or something else?
You're talking about two different things. The pain you experience during or right after the exercise is called acute muscle soreness:
Acute muscle soreness is the pain felt in muscles during and immediately after strenuous physical exercise. The pain appears within a minute of contracting the muscle and it will disappear within two or three minutes or up to several hours after relaxing it.
The following causes have been proposed for acute muscle soreness:
Accumulation of chemical end products of exercise in muscle cells, such as H+
Tissue edema caused by the shifting of blood plasma into the muscle tissue during contraction
Muscle fatigue (the muscle tires and cannot contract any more)
The pain that takes about 24 hours after exercise to show up is called delayed onset muscle soreness, or DOMS:
Delayed onset muscle soreness (DOMS) is the pain and stiffness felt in muscles several hours to days after unaccustomed or strenuous exercise.
The soreness is felt most strongly 24 to 72 hours after the exercise. It is thought to be caused by eccentric (lengthening) exercise, which causes small-scale damage (microtrauma) to the muscle fibers. After such exercise, the muscle adapts rapidly to prevent muscle damage, and thereby soreness, if the exercise is repeated.
Minor pain that doesn't hinder your daily activities is normal. Beyond that, it's not.