How does the system know how to read the correct block from a disk? I am watching this video (https://www.youtube.com/watch?v=aZjYr87r1b8) and up to the ~6 minute mark, he explains the disk structure and how when you want to read a block, the spinner spins to the correct sector and the arm to the correct track. How do the spinner know where the correct sector is? How does the arm know where the correct track is? And how do they know they are reading from the beginning of the block and not in the middle of it?
Finding the right track is relatively easy. There is a motor which can be moved a precise distance. Then the head can measure magnetism to the right and the left and use this to move to the precise centre of the track. 

The sectors all have a header. The read head passes sectors at high  speed. The start of the header has a distinct pattern that cannot be found inside a sector. So the read head can wait until it finds the start of a sector. The header also contains a number identifying the sector so the read head knows when it reaches the correct sector and can start reading. 

In practice, all sectors on the track will be read in case another sector is needed soon after.