Quarter Circle Motions and Input Display
Training mode offers a very helpful feature called input display that shows every individual input you make. This is incredibly helpful to learn quarter circle motions (a staple for special moves and supers) or diagnose input errors. As for quarter circle motions the aim is to start with your thumb in the down direction and to roll it to the direction you want.
The game engine defines it as Down, Down-Forward and then Forward. This is hard to get used to at first, follow the below steps to make this motion second nature.
Jump into training mode with Goku and turn on input display in the start menu
Practice the motion without pressing any attack buttons. With input display on, you should clearly see three arrows. Down, Down-forward and Forward (236 in proper notation).
If you only see Down, Down-forward then you need to make sure you are lifting your thumb up enough so that down is not held
If you only see Down and then forward, try to consciously slow the motion down and make sure you press both down and forward at the same time.
Once you see these arrows correctly in sequence, practice it until it looks right every time.
Now it's time to include the "Special" button. Perform the quarter circle as fast as you can after you let go of down and are only holding forward. You should fire off a kamehameha.
If it still is not coming out, consult your input display once more, if you see Down, Down-forward, Special, Forward then you need to hit the button later. If you motion looks correct and is followed by the Special input then you must be pressing the button a bit too late. Keep going and keep adjusting your timing until your input looks correct and your special is coming out.
Once you can do this 10 times in a row, jump over your opponent so they are on the left side and repeat the previous steps until you can do it 10 times in a row.