oh boy, this gets tricky to explain. There are a few types of mixers, one just spits out the two signals that you put in at the same time with no notable change to either signal. You don't want this type. You want the mixer that will act as a math function where you produce a sum and difference of the two signals that you input. With this type, you have a rich mixtures of signal components to play with. You can filter anywhere along this spectrum to produce all sorts of sounds. You want your mixer to be of the non-linear type (Not sure how to build one for Audio), not the type that just combine to audio signals into an opamp.
For making your various waveforms. Revisit the link I gave you earlier on the XR2206. That will produce square, sine, triangle, and a few others.