This may seem like a silly question, but I've just managed to change the chain and cassette on my bike, and now I have an old chain and cassette to get rid of. Is it generally okay to just throw this stuff in with the regular garbage? Or should I take it down to the DPW? I suppose I could look on the local DPW website and see if it says anything about bike parts, but wondering what you all do.

Throw it in the recycle bin - most bike parts are recyclable metals.

If you have a recycling point for tins in your area and you regularly save and dispose of your tins and other small metal objects then just put the chain and cassette into the bin along with the cans and such. If you regularly toss your cans and other small metal things into the general trash then that's where you can put your bike parts.

i give my broken parts to a local guy that does scrap. i toss all the parts in a recycling bin and put it to the curb the day after garbage day and he comes and picks it up.

If your city has recycling, ask if they want it in with the cans and such, or what to do otherwise. You might also call the local boy scout troup, Many raise funds by recycling metal, though they want aluminum much more than steel.

If there's an auto graveyard anywhere nearby and you've a good throwing arm, pitch them onto the pile of car bodies and it'll get to the right place.

The recycling is a good idea--these are basic steel bits, so that should be a problem. They'll take it as scrap metal at the drop-off center (basically a bin at the DPW). I got there with food waste for composting, so no extra work involved, really.

^^^
That way one of us cheap curmudgeons will fish it out of the bin later and crow about out dump find!!