Do I need an oscilloscope?

No. For a few troubleshooting issues, having an oscilloscope available is useful. But you would also need the training to know how to use one (not difficult, but not trivial). If you enjoy amp building so much that you want to get into the real detail of waveforms, distortion, etc, then it might be worth the initial expense and time spent educating yourself.

OK – how about a signal generator?
Again, this is not required, but is potentially useful as a companion gadget to use with the oscilloscope. For example, you can feed a 100mV sine wave into the amplifier’s input and then see how it is amplified and distorted as it travels through the amplifier to the speaker output. (Warning: your Other Half will voice some displeasure at 18W of 1kHz sine wave tone piercing its way through the house and/or neighbourhood. You have been warned!)

OK – how about one of those fancy spectrum analysers?
Very useful for experimenting, but only worthwhile if you’ve won the lottery!