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.
Regarding the capacitor, Ignacio beat me to it. shakefist
You'll see capacitive coupling all over the place in circuits like these.
As far as what you feed in, actual amplification will depend on the resistor values. The specs say -.3 to vdd (your source voltage) + .3V, so provided your VDD is >= 3 volts, you should be able to amplify to that level.
update
You can simply leave the V01 disconnected. The speaker wire connected to pin 5 in that schematic would be connected to ground instead.
That said, the configuration shown in the schematic is said to have a gain of $$2(\frac{R_i}{R_f})$$ This is because the outputs are 180 degrees out of phase with eachother, so that even though both amplifiers being unity gain (when Ri = Rf), it winds up doubling the input.
If you don't use pin 5(V01), you'll need to adjust your gain. The new formula would be $$\frac{R_i}{R_f}$$
That said, if you haven't already ordered a different chip, I think the LM4880 is more in line with what you want, it's mentioned in the automatic switching circuit in the datasheet, and it's a single output headphone amplifier.
Best Answer
Part 1 - Use 2N3055 to learn audio amplifier
Update 2020sep24hkt14:47
How nice to see @ocrdu updating our crowded answer with the following critical jigsaw puzzle piece!
Update and Errata 2020sep24hkt1322 - The above title before this update had a typo - 2N3005 SCR should read 2N3005 NPN transistor. Many thanks to @Andrew Morton for pointing out my severe careless mistake.
2SC1050 is not that popular, therefore you find it difficult to google things for it.
Instead you may like to try 2N3055, which can do what 2SC1050 can, and is perhaps 100 times more popular, so you can find tons of tutorials on it.
The following is an example of a couple of YouTube tutorials to get you started.
(1) DIY Audio Amplifier with 2N3055 Transistor - cbz 2019nov23
Part 2 Small Power audio amplifiers
2SC2050 or 2N3055 are for big power audio amp. For smaller power, you might like to consider the following.
(a) DIY PAM8610 Digital HiFi Power Amplifier Board 15W + 15W Small Power Mini Audio Amp Kit DC12V - US$8
(b) LM386 Low Voltage Audio Power Amplifier - TI
References
(1) Whatever Happened To The Electronics Hobbyist?
(2) eevBLAB #2 - Are Electronics Hobbyists Useless? (YouTube)
(3) EEVblog #874 - World's Oldest Electronics Hobbyist! (YouTube)
(4) The Decline of Hobby Electronics? (YouTube)
(5) An Electronics Workspace for a Design Engineer - elektormagazine
(6) The 10 Best Raspberry Pi Stores Available in the Market
(7) Inside a 12W LED module (12:26 Schematic) - BigClive
(8) 2SC1050 Based Power Amplifier
(9) JohnAudioTech Videos