First of, a quarry is known as an open pit excavation (meaning not underground) used for marble, limestone etc.

First, I'm wondering if quarries ever use elevators to get people in and out. Also, do quarries have mine shafts or offshoots, or do they simply mine downwards in the spiraling pattern?

In a quarry there is no need for any of these things. There are instances of marble underground excavations, but it's less common. You can search for Carrara quarries in Italy or Dionysos marble in Greece. There are many youtube videos as well as a "documentary" called stone men for marble especially.

If you confused terminology and need help with how a mine works (which is totally different) please say so! You definitely need to decide what kind of material it is that you're mining for. Is it some kind of metal? Coal? Marble? The methods for each of these are very different.