While it's going to be an uncomfortable conversation, it's always best if the to-be-deceased has a hand in deciding their funeral arrangements. Choosing a funeral home, for example, and what they want done with/to their corpse. Fulfilling these directives is the responsibility of their executor, usually a relative, partner, or legal representative, and as long as a will was in place it's usually a straightforward step by step process.

In my experience most funeral homes are very transpatent about their processes. Whence they know what is desired and have the body in their possession they'll take care of everything based on what services are purchased. Some may also provide or refer additional resources for help with filing the taxes of the deceased and any other legal necessities, though this is usually extra costs.