Best transistor to use for audio amplifier
You could successfully build a audio amp from many different types of BJTs.  It will be the circuit, not the transistor, that makes the amp work well.  I'd pick jellybean parts like the 2N4401 (NPN) and 2N4403 (PNP) and stick with them for everything except for the final power output transistors.  Lots of parts could fill that role.  If you have your own favorite jellybean small signal transistors, use them if you prefer.  The ones I mentioned have reasonable gain and can handle up to 40 V, which should be plenty good enough to allow for a amp to impress your profesor with.

There are lots of possible power transistors to use as the final output.  If you are aiming for a few Watts, I'd probably go with basic parts like the TIP41 (NPN) and TIP42 (PNP).

Again though, it's not the choice of transistor that will make or break this project.  You can certainly create a impressive audio amp with the transistors I mention, but you can also make a mess.  It's really up to the design.  In audio, overall noise and harmonic distortion are high priorities.  Those come from careful circuit design and attention to these parameters at every step along the way.

You can also use other types of transistors, like JFETs or MOSFETs.  Those would require a different circuit topology to utilize properly, but can be used to make a good amp too.  Since you will be going over BJT details more thoroughly, I'd stick to them for now.  This will be a great learning exercise.  Designing a amp with very low noise and very low distortion is not trivial.