How does one approach designing an amplifier using BJTs to drive an 8 ohm speaker?
You are asking way too many basic questions for this stage in your course.  This means you didn't understand some previous material, but worse, it's clear you don't have any intuition and feel for circuits.  With some hard work you can possibly muddle your way thru and eventually receive a degree, but that will be just a hollow piece of paper.  I've encountered such "engineers" a few times, and they are outed quickly in the real world.  If you haven't been tinkering with this stuff since at least high school, then you don't have a passion for it and should seriously re-evaluate your career objectives.

However, to answer some of your question, break the problem into three stages.

The first stage should have a reasonably high input impedance with the ability to receive the feedback signal from the output.  This could be, as only one example, a differential pair as found on the input of BJT opamps.  The main purpose of this stage is to take the input signal, subtract the feedback signal, and multiply by some gain.  It should also lower the impedance of the signal that it passes to the next stage.

The second stage is largely for gain and to make the signal even lower impedance.  At this point it's just a single ended signal.  There are many ways to do this.

The third stage is to provide the relatively large current needed to drive the speaker.  This stage won't have much voltage gain.  In fact in some configurations it actually has a voltage gain less than 1.  However, it has significant power gain due to transforming the impedance down to something small compared to the 8 &Omega; of the speaker.  Hint 1: A BJT in emitter follower configuration has voltage gain a little less than 1, but large current gain.  One problem is that the output voltage is the input voltage minus the B-E drop.  Hint 2: Ordinary silicon diodes have about the same voltage drop across them when forward biased as the B-E junction of a transistor.  With a little cleverness, you can use a diode to offset the input of a emitter follower to get roughly the same output level as the input level, but still with the large current gain.  Hint 3: Both NPN and PNP can be used as emitter followers, each working in the opposite polarity.  Both together can be used to make a emitter follower that works symmetrically in both directions.