Producing different types of tones using the 555 timer
First, if you are trying to make something musically interesting with the 555 timer, your best bet is to have a look at the Atari Punk Console.

On to the questions: a 555 timer actually produces two easily accessible waveforms: a square wave and a pseudo-triangle wave. Let's look at the astable circuit:







simulate this circuit &ndash; Schematic created using CircuitLab

Running a transient simulation of the circuit gives us the following:

The pseudo-triangular waveform observed here is from filtering of the square-wave to produce a more "pure" tone, removing the high frequency content of the square wave and moving it closer to a sine wave. In musical synthesis this is referred to as subtractive synthesis as it removes content from the signal to produce the output, here achieved by the low pass filter from the RC subcircuit. (Note this can't be directly listened to without buffering because changing the impedance ruins the oscillator).

To get to a sine wave, you would have to have a high-order filter just above the fundamental of the square-wave to remove all harmonics. This would be challenging to do for a synth as the fundamental changes and therefore the filter would have to change cutoff frequency as well, which is challenging to design. 


A direct answer to your question on how to get a sine wave would be to stack several RC stages in series and that would filter the output enough to produce something close to a sine. You may need a buffer (e.g. from an op-amp) first.








simulate this circuit

The above schematic gives the following output:

This is not a perfect sine wave but is much closer than the "triangle"! The circuit is three stages of a low pass RC filter forming a 3rd-order filter. To calculate the cutoff of the RC filter you simply use
$$
f_\mathrm{c} = \frac{1}{2 \pi R C}
$$
For a cutoff of 132 Hz you could use a 10n capacitor with a 120k resistor, but you can tweak the values to what you have available and how accurate you need it.


To create more complex signals you need more sophisticated synthesis techniques. These are often easier to implement in software as to generate the waveform you reference one could simply draw it into a wavetable synthesiser. To create that exact waveform in hardware (without a microcontroller or similar) would be difficult but could be done with analysis of the frequency response and some approximation using an appropriate synthesis technique (e.g. amplitude modulation, frequency modulation).


Finally, if I were you and were trying to build a basic synthesiser using 555 timers, I would look at combining oscillators and filters. By taking a single 555 and controlling the pitch (continuously) and mixing it with a second oscillator you can create some weird sounds. Look up how to make a buffer and a summing amplifier with an op-amp and you will be able to make a huge amount of different sounds.

Good luck!