Etching Silicon Dioxide
SiO2 can be wet etched in a home chip lab environment easily with ~2% HF from Whink Rust Stain remover, which can be purchased on amazon. It can take 15+ minutes at times to etch all the way through a field oxide at room temperature, but can be greatly sped up if you heat the etch solution. Also note that the etch solutions have a very short life time due to the low HF content, you will usually see small bubbles in an active HF etch solution during the first few etch cycles and will not be present after that. Then you will notice that the etch rate greatly decreases and then stops, at which point you will need to get more Whink. It should also go without saying that extreme care should be exercised while working with HF, including all PPE, thick gloves, mask, gown, etc. You should also have HF skin treatment cream nearby, such as Calgonate .
Etching Silicon Nitride
Silicon Nitride can be etched with hot concentrated Phosphoric Acid. Read more about this here .
Etching Bulk Silicon (MEMS, etc)
KOH and TMAH can be used to etch silicon surfaces with reasonable rates. Read more about this here .
Commonly, exposed and developed photoresists are used as etch masks to pattern wafers but for a home chip lab, electrical tape or adhesive vinyl sign material can be used as a quick mask to make diodes and transistors. The vinyl sheets can be cut by hand with a knife to make individual transistors or on a laser for more complicated or smaller designs.