If our minimum wage goes up, doesn't our cost of living go up as well ?

Yes the cost of living will go up as well.

If a minimum wage increase is implemented over a period of a few years then the economy has time to adjust to it.

So in that case the negative effects are not as great.

But if you attempt to practically double the minimum wage overnight then you will cause a severe reaction throughout the economy.

The retail and restaurant industries would be heavily affected.

Many of them would be forced to lay off as much as half of their staff.

At this point there is not much room to raise prices as the customers can not afford the increases due to massive unemployment from the Covid shut downs.

And if they lower the quality of the product too much they will loose customers.

Plus the argument of the minimum wage is not a living wage is a fallacy.

It was never meant to be a living wage.

Minimum wage is supposed to be the wage a new employee with no experience would get when starting on a job.

If they do a good job they get these things we call raises and promotions.

If they decide to stay in retail or the restaurant industry they move on to management or to better establishments that pay better and they get better tips.

But many of us moved on to different jobs entirely.

I earned minimum wage but withing a couple months my performance on the job was god enough to get a raise.

Many jobs that you will get minimum wage advertise that you will get a raise with in three months.

If your performance is not good enough to gain a raise they will have already fired you.

Not necessarily. Some companies can afford to simply pay their shareholders less dividend, or to postpone purchases/investments

Everything will go up too.  Plus, most employers just can not pay that much - especially for minimum wage jobs.  Employers will be forced to cut employee hours, lay employees off and close their doors.

Not necessarily. 

If minimum wage goes up to $15, a lot of business will layoff people, unless they can raise their prices. If we print more money, then we get inflation and prices will rise anyway because of that. $15 is not reasonable. It should be between $7.25 and $15, like $10.