I’m currently playing classical and pop (marching) music and am interested in upgrading my mouthpiece (4C, there’s a lot of room for improvement). I for sure plan on continuing with the marching, but I still want a mouthpiece usable for classical playing, as I might pursue that as well.

I’m planning on spending under $100, and have no clue about mouthpiece quality.

Also, what effect do chamber and facing have on sound?

You’ll want to get a Selmer mouthpiece, ask about specific models at the store. Facing affects the gap between the reed and the tip of the mouthpiece. With larger facings, you can push more air through the sax and be louder, but with smaller ones, there is better tone and control.