I've looked for tutorials that can teach me to use multiple push-buttons at the same time, with no luck. What is the best way (or at least a good start) to connect 15 or more buttons that I can use, and possibly use about half of them at once?
I know this is a vague question, but surely someone's done this before and can help, even if it's only a little bit.
What exactly do you mean by "at once" - it probably has a very different meaning for an Arduino compared to a human because humans do things slowly.
If you need to check that certain buttons have been pressed within a certain time you need to save the value of millis() when the first button is pressed and check what buttons have been pressed when the time expires.
You have up to 7 fingers holding down 7 buttons? That's a lot.
A basic row&column "matrix" keyboard can reliably detect two buttons held down and 3 or more with some limitations.
If you add a diode to each button then the matrix will work better for multiple presses because current can't flow back through the buttons the 'wrong way'.
How far apart are these buttons? If they are controlled by separate people then I'd probably have a small Arduino control the small group of buttons that are in front of each person.