The first 48–72 hours

Right now.

Get a legal pronouncement of death (hospital/hospice handles it; if at home, call your hospice line or local non-emergency line).
Honor anything time-sensitive — organ/eye/tissue donation has a short window, as do some religious or cultural rites.
Tell the people closest to them; ask one or two people to help spread the word.
Secure their home — lock up, bring in the mail, set heat/lights, protect valuables.
Care for dependents and pets right away.
Arrange care of the body; a funeral home or cremation provider can move them before you decide on services, and you're entitled to itemized prices first.
If it's easy, note where key documents might be (will, trust, life insurance) — just check the obvious places.

The first 1–2 weeks

The early decisions.

Plan the funeral, memorial, or celebration of life — first check for prepaid arrangements or written wishes.
Order the death certificate, and get more certified copies than you think (10–15); nearly every institution wants an original.
Share an obituary or notice if you'd like one.
Notify their employer; ask about final pay, benefits, and any life insurance through work.
Notify Social Security to stop/redirect benefits and ask about survivor benefits.
Identify the executor or personal representative named in the will (or who will step forward if there isn't one).
Start one simple log — who you've notified, account numbers, passwords you find, and open to-dos.

Settling the essentials.

Meet with an estate attorney to determine the path: trust administration vs. probate (very different processes).
Notify banks, brokerages, retirement plans, and pensions — note that beneficiary designations often pass outside the will.
File life insurance claims, including easy-to-forget ones (work, mortgage, credit card, associations).
Cancel or transfer credit cards, loans, utilities, subscriptions, memberships, and phone service.
Guard against fraud — notify the three credit bureaus to flag the file as deceased.
Handle real property — mortgage servicer, homeowners insurer (vacant-home coverage may be needed), property taxes; start thinking keep/rent/sell.
Deal with vehicles, safe-deposit boxes, and other titled assets.
Notify other agencies as they apply — VA (benefits/burial), DMV, voter registration.

Closing things out.

Open the estate and begin probate or trust administration; inventory assets and debts.
File the final personal income tax return, and an estate return if required; a tax pro is worth it (basis step-up on inherited assets matters, especially for property).
Pay valid debts, then distribute assets per the will or trust.
Close or memorialize digital accounts (email, social, cloud storage, photos, crypto).
Keep copies of every filing, statement, and decision.