Start small, and take one step at a time! A good starting point is to learn to generate and play notes with well defined timing and pauses. For decent audio volume you will need an audio amplifier and speaker.
For decent tone quality, you will need a low pass filter between the Arduino PWM output and the audio amplifier. A two stage RC filter with a rolloff at about 2 kHz is a reasonable choice.
If you want to play more than one tone at a time, learn about polyphonic synthesis.
Add one new feature at a time, testing as you go.