So let's imagine minimum wage went up to 15 dollars an hour overnight.
1) There would be layoffs in the morning for a lot of businesses. Outsourcing etc.
2) You'd probably see restructuring of various industries rather quickly. At least those that employ minimum wage workers. Perhaps fast-food has more self order touch screens or something along those lines. Clothing stores, already hurting, would probably shut down quite a few stores. Online shopping would become even more common.
3) You'd see an increase in motivation to find work. Competition for basic level jobs would increase. In fact, sort of like more selective colleges, businesses would demand more from perspective applicants to hire them, even for basic jobs.
4) Cost of products would go up.
5) People would have more money at least in raw terms.
I guess, I sort of feel like at least in the immediate future, it would probably hurt the untrained quite a bit. Their jobs would either be outsourced, taken away, redesigned etc. They would probably have more trouble getting hired. These are the people most strongly advocating the increase too, imo.
Therefore, a more moderate increase would probably be better. So something like 10 or so. Not enough to incentivize trained to take untrained jobs, but enough to help untrained.
In general, the divide between lowest earners and highest earners is too great.
So a combination of raising the minimum wage, more direct(no loopholes) on taxation of the wealthy, and cutting of some benefits is needed. What percentage those things are changed by depends on your politics.