I think a lot of people our age feel the same way towards money. We wish we could reject it but we can't. It's been imprinted on our brains since birth. Maybe that is why instant india is very popular among our age bracket, because in following the eastern practices we think we might actually reject the west's money culture. But we can't.

Rejecting money would mean you should turn off the screen in front of you, throw your phone in the ocean along with your education and go live in the woods. There's nowhere you can go to escape money. Even monks need to buy some things. And there is nothing wrong with that. There is nothing wrong with saving money and buying comfort.

The problem you're getting at is the danger that money becomes the ultimate goal. But as you are writing about your worries I can tell you would have to become quite deluded to let money win the day. Don't be afraid to spend money. Buy something that you WANT and don't need every once in awhile. It will remind you that money is just a game. It grows and it shrinks and you can exchange it for prizes. Get prizes and give prizes. Give as much money away as you feasibly can. It's such a nice feeling to let others enjoy the fruits of your labor. (Remember always to offer your guests fruit)

And right now, as you are my age, my only advice is to live cheaply and forget money. Your focus should be on your future and not the results from your future. Take jobs that pay nothing as long as they are what you want to do. I know so many recent college graduates that are scared to death. They have no jobs and no prospects and they are afraid they won't be able to live. But it's not hard to get by, and don't worry that you are only getting by, money will come by itself as long as you never pick a job based on it.

You ask how you'll know when money becomes all important. If you ever find yourself at a job only because it's "safe and steady money" then you'll know. At that point you should quit and find a job that might not be safe and steady money, but is at least fun enough that it will distract you from how poorly you are living ha.

To reject money all you have to do is not take it seriously:

1) Laugh about it sometimes
2) Treat yourself sometimes
3) Give it away sometimes
4) Leave it at home sometimes

Let's compare money and sex for a moment. People should want to have sex in order to enjoy it, or procreate, etc. (see discussion on what's the point of sex), but much of people's obsession with sex is only to have it for the sake of having had it. So they can tell others they had it, or even tell themselves (who wants to be a virgin after a certain age). So they can feel good about themselves. The sex itself is, sadly, not at all the point for most people.
Similarly with money. Money is supposedly a tool, but it has turned to be the goal itself, perhaps since always. People want to have money for money's sake and feel their own worth by the amount of money they have or can earn.
For example, I know someone who came from a very rich family and didn't need to work to gain a living. He was well considered as a philosophy graduate student, but only when he started working and earned his own money did he feel he was worth something. It didn't matter what his professors said, it didn't matter that he didn't need the money at all for anything, but earning money, that's how he felt he was worth something in the world. I know several cases like him.

The value of money, for most people, is purely symbolic, and not at all utilitarian. Try to have money merely be utilitarian. Money is not worth anything more than what you can buy with it.

I don’t hate money because we need it. It helps us live with dignity and to fulfill our needs and wishes. All I can say is if you want money to achieve a goal, to have comfort, or even for a stylish lifestyle, it bears no harm. The danger with money is to fall in love with the money itself and not with the things it helps us achieve.

It is very hard to study, to raise children, to open one’s mind to culture, art, and to the needs of other people without a basic amount of money. Without it, life is very hard and it could make you act and behave in a way that you don’t like. 
Use the money enjoy it but don't fall in love with it, then you know it won't harm you and your surroundings

Money can make you happy if you know how you choose to live.