how do I manipulate individual bits within a byte?
You can do it the grown up way using and and or operations. On the other hand the Arduino offers a beginners bit read, bit set and bit clear operation.
How do I display each bit within a byte,
A simple loop printing out a 1 or 0 depending on the value of the least significant bit. Then next time round the loop shift the byte to the right.
Again the and operation will help you isolate the least significant bit or you could use the bit read.