Health Care

You mentioned "a long stay" in Italy. Note that the "Schengen rules" may impose strictly enforced legal limits on the length of your stay in Italy and Europe, unless you're exempt depending on your personal nationality status.
You'll probably want to research this issue. It's been discussed here a lot, use the Search box at top of this window to find previous discussions.

The application for a long term stay visa will spell out what is required. Then contact health insurance companies to see if they can fulfill the requirements.

If by long stay you mean you want to seek a visa and then a residency status, you will need proof of private health insurance (not Medicare). If by long stay you mean a month or two, you should check with your insurer if you have one or buy a travel policy.

It would help to have a bit more information. What length of "long stay" are you considering? There are people on the forum that have resided in, or are currently residing in Italy, so I'm sure they will be able to provide lots of good information.

For up to 90 days, there are several travel insurance policies, like TravelGuard, that can provide coverage for health emergencies.

For longer than 90 days, you need a long term visa, such as an Elective Residency visa, unless you qualify under other long term visas (e.g. study abroad). Information is provided in any of the Italian Consulates’ websites (below I linked the one in San Francisco).
https://conssanfrancisco.esteri.it/consolato_sanfrancisco/en/i_servizi/per_i_cittadini/visas.html

For a long term visa you need proof of insurance. Besides private insurance, you could also buy into the Italian National Health Service. In this latter case you can only buy coverage for one full year at once (1/1 through 12/31) and cannot be divided for portions shorter than the full year.
http://www.salute.gov.it/imgs/C_17_opuscoliPoster_118_ulterioriallegati_ulterioreallegato_4_alleg.pdf

The cost of purchasing coverage for one year from the Italian NHS depends on your income earned (in Italy or abroad) during the preceding calendar year.
The annual cost is calculated as follows:
For income portion up to €20,658.28 > 7.5% of annual income.
For income portion between €20,658.28 and the max ceiling of €51,645.69 > 4% of income
Therefore the max annual cost is €2,788.87 (which, if paid for the full 12 months, works out to about €232, or $260 a month, a bargain by American standards)
Students and Au pairs pay much less.
http://www.salute.gov.it/imgs/C_17_pagineAree_2522_listaFile_itemName_0_file.pdf

I use American Express when I buy health coverage for overseas as well as medical evacuation.

Wow, Roberto, no soaking of the rich in the Italian Healthcare System (unlike US Medicare, Part B that charges almost everyone at least $135 and goes up from there based on your tax return from 2 years ago when you were still working and doesn't tell you that until you've already opted out of your spouse's company insurance because you didn't like the company plan's choice of doctors.)

cala
The figures I gave are for foreigners on Elective Residency (basically non working retirees) voluntarily purchasing into the system. Foreign students pay almost nothing (less than 150€ a year).
Italians and foreign permanent residents (in the country to work) don’t have to pay anything. They are automatically enrolled and don’t have to pay anything above what they already owe in taxes based on their income. The system is funded by a variety of taxes (Payroll taxes, VAT, income tax).
If you can get to the country as a relative of a citizen or perm resident (like a spouse) you also pay nothing extra.

Cala, take your Federal tax returns proving lower post-retirement income to a Social Security office and ask for an adjustment. The older return is just a default based on the latest information available to them. I know of situations where the premium was reduced back to the beginning of the coverage year. It's worth a shot if you're being charged way more than the $135.

And if you have part D prescription coverage, that could go down as well.