If you're creating a square wave with a PIC, it'll sound buzzy. That's
what square waves sound like. To change the tone you'll have to
change the shape of the wave generated or pass it through a tone
control or shaping circuit.

A triangular wave is reasonably close to a sinusoidal wave and is
quite a pleasant sound. It would be easier to generate that from a
DAC than a sine wave, especially for different notes. A triangular
wave passed through a simple bass & treble tone control would
be pretty good. The tone control need not be an active one - an RC
type would do, but you may have to compensate for lost signal level
with some amplification. An op-amp or transistor mixer could do that.
And an active TC doesn't necessarily need front panel pots, it can
be done with cheap pre-sets.

If you want to mix square wave frequencies, put each through its
own tone control before you mix them. You could try to get at least
some linear (ie not just 0V or 5V) values to reduce the harshness
of the sound. Mixing frequencies means you'll end up with chords
of some description, no problem to run those through a single
speaker, otherwise we'd all have stereos with truck-sized speaker
cabinets.

If you haven't got a DAC-capable PIC, then you could use an
external one to produce non-square waves. Another option could
be to make your own with resistors and/or digital switches.

A quad op-amp (LM324) could be used for 3 active tone controls
and a mixer. A transistor on the o/p of the mixer could give a bit
more ooomph to drive a small speaker. There'll be plenty of practical
circuits out there to suit any application you're thinking of.

Jinx