Singing is a surprisingly complex thing that requires many things to work well together, it requires you to actively use very specific musculature and develop feeling for your body. Bad habits in singing can lead to actual damage to your voice.
So if you want to get into singing it would be a really good idea to get some supervision, i.e. a teacher. Even if you cannot take lessons on a weeks basis – you will profit from regular lessons, even if it is just once a month.
Generally it is good to warm up before singing, and warming up should consist of
- Getting into a good posture and tension (not too relaxed, not too tense), stretching, such things
- Reminding you brain how specifically your vocal musculature works
You can extend warmup to train technique (rather than literature, where you need to focus on so many other things).
And singing a lot is also important to build musculature and training your body feeling and brain into getting what you want.
But this can only really be effective if you know what to look for or if you have feedback how to improve. And this is why a teacher is so important, especially in the beginning (of course there are always people who just do things right by themselves).
From the recording you’ve given us: Your voice has a lot of air. This is caused by the vocal cords not being closed fully, and it will mean you are losing lots of air for no effect. Once you learn how to property set your vocal muscles you will be able to produce more volume and longer phrases.
One approach to get into singing would be to join a suitable amateur choir. A good choir conductor will do good warmup and combine it with exercises for vocal training. But still the most efficient way would be to get lessons now or then.