How or where do I get rid of an old mattress? If I buy a new one, will the furniture store take the old one?

Mattress dealers often arrange to take away the old one when they deliver the new product.

If not, arrangements can be made with your household waste hauler [whether municipal or private] to take it, usually for a nominal fee.

Some do, yes. You would want to contact local businesses and buy from one who has that policy, or pay the dump to take it.

If it's in usable condition, you can try selling it.

Are we supposed to guess where you live?

no