If you have a job, any job, be it part time or full time, and you are not getting enough money to make ends meet, ask for a raise in income, if that does not work, look for a better paying job.
How comes some people just can't see it that way?
Lets all boycott McDonald's because they don't pay enough money per hour
Consider it this way:
Someone is earning $10/hour, that's more than the $7.25 federal minimum wage. On a national average $15/hour is considered roughly to be a living wage. Do the math on a living wage for a totally independent person who is paying 100% of their way in the world. No subsidies of any kind. $15/hour may sound like a lot but it's not really enough.
You are proposing that the person looking for a living wage will increase their income by 50% ($10->$15). 50% raises are not likely. You are assuming they have improvable skills. Well, they can take their low IQ/bad education/learning disabilities/whatever and try, but there are limits. I'm not ever going to be a rocket scientist, for example. We all have limits, some higher, some lower than others.
If the market values that person's worth at $10/hour, who will pay more? They may even want to, but can't because they are competing with others who are paying $10/hour, or China paying $3/hour. Or, they are locked into a franchise deal that has no room in it for $15/hour wages. In some regards, it's a Mexican Standoff.
If a living wage floor is established at $15/hour, then the playing field is leveled among competitors as to how cheap labor can be. They all pay a minimum living wage, they are all free to continue competing in every way but minimum labor rates. The existing minimum wage works like that, there are exceptions, workers who receive tips, for example.
$15/hour is low enough that most people will try to earn more, and many will need to in any case. They are free to use their skills to compete for more, and they will. They are human after all.
How is that different than what happens now? Now we pay lots of people less than a living wage, then we compensate for that with subsidies. Then we complain about the cost of subsidies (taxes), and the administrative overhead (government) to manage those subsidies. How is that an improvement over paying a living wage to begin with? It's actually more expensive.
Even in the ideal scenario where a person works themselves up to higher wages, the time that they spent earning less than a living wage usually incurs welfare costs of one kind or another. I don't think that makes a lot of sense, but it would be a typical rags to Carhartt American success story.
The absolute reality is that without a minimum wage floor
and the expensive subsidies we now have, there would be many homeless and starving people. The cost of living simply exceeds their earning ability. If we eliminate subsidies and minimum wages, we are not valuing them as people, but rather only on what they are worth in the market--like a commodity. People and "commodity people" are not equivalent social values.