You would be forgiven for believing that to sing long notes you need to spend hours practicing breathing exercises, but that is only true to some extent. While breathing exercises are good for relaxation, to hold long notes, we combine good breath control with good pitching and control of the diaphragm.
There are some teachers who don't like to focus on breathing and diaphragm. I must point out here that I completely understand why. However, in my experience, specialising in working with nervous singers and individuals with anxiety, using the diaphragm properly can release a lot of tension. Releasing tension allows better breath control more quickly, and less shaking, leading to more accurate pitching.
An effective method I use is the candle method.
Take a normal household candle (don't use your housemate's favourite scented candle!) Take a piece of card or folded piece of A4. Make a hole in the centre and poke the candle through. Hold it from the bottom, underneath the card or paper (the card will catch the wax).
Hold the candle a few inches in front of you (away from your hair!). Breathe into your ribs for a count of four. Hiss the breath out for a count of eight, without extinguishing the flame or making it flicker. DO this a few times until you feel the breath leaving smoothly.
Extinguish the flame. Breathe in again for a count of four, adding a hum. When you feel ready to move on, add a vowel.
Over time, build up to count of twelve, then sixteen and so on.
It's not compulsory to use a candle but it's great for a visual and for tactile people!
Children should, of course, be supervised by a responsible adult.