Phone Rental

Just buy a cheap phone for max 39 euros (in France, go to Leclerc) in Italy, you will find shops everywhere selling cheap phones.
You will already get a SIM card with the phone you buy. ( either a French or Italian SIM card)
With it, you ask to charge it with a card. These cards have several possible amounts and you have to follow what is written on the card. In Italy, you can buy these cards at any Tabaccaio.
In France, depending on the company, you can buy it at Leclerc, Tabac, post office, orange shops, SFR, Bouygues telecom.
Buying a SIM card is cheap.

Robert,

If you only need talk and text, you don't need a "smartphone". A basic phone will work just fine.

Renting phones tends to be more expensive than just buying a basic model. You might have a look at the travel phone firms such as iRoam, Cellular Abroad, Telestial or Mobal (among others). They can provide phones for as low as $29 with SIM cards that will work in the countries you're visiting with reasonably consistent rates. Some may also rent phones.

As a "Luddite" you may also want to familiarize yourself with European dialing methods.....

https://www.ricksteves.com/travel-tips/phones-tech/call-uk-europe

To get the best specific advice for your situation, tell us more about what you intend to do with the phone. Will you be making calls within the country (say, to a hotel or restaurant)? Will you be receiving calls from that country? Will you be communicating with travel partners, and if so, what phones will they have (French ones, US ones, etc?). Will you be calling back to the US? Will people in the US be calling you? Will it be for emergencies only, or do you intend to use it a lot?

These days, phone rentals are both hard to find and expensive. You will want to buy a phone. And as Ken said, you don't want or need an expensive smartphone. In fact, ask around among your friends if someone has an old, unlocked, quadband "dumbphone" that they're no longer using. You may be able to get a suitable phone from them for free. However, it must be both quadband and unlocked for it to work in Europe.

As a starter, read Rick's post about phone basics: https://www.ricksteves.com/travel-tips/phones-tech/cell-phone-europe

There are two kinds of SIM cards. Every time you get a new SIM you get a new phone number. If you're the one making the calls, this is not a problem. If you're receiving calls (particularly if the reason you're getting the phone is that you need to be reachable during your trip), it means you have to let everyone know the new number (you can text or e-mail them).

One kind of SIM is "international," like the ones Ken listed. These work in multiple countries at fixed rates, and you get to keep the same number forever, and they're easy to top up, but they have higher rates. While your number doesn't change, it's usually a UK one (some are tied to other countries), so it's not a local call for someone in France or Italy. But you can keep the SIM from trip to trip, and you can keep the phone with its SIM next to your passport, ready for any trip. They will even work in the US (sometimes high rates), so it works for emergencies here too.

The other kind is "national," say from a French company like Orange or an Italian company like TIM. These are cheap in the country of origin, and within the EU the rates are capped. They can be hard to top up outside the country of purchase (in theory you should be able to do this online, but many have had problems). And the SIM expires after a certain interval of not being used or topped up (particularly in France, this interval is often short). So, you usually can't keep the same SIM from trip to trip. But you will get the best rates, and having a local number is great if you're receiving calls from that country. If you're communicating with travel partners, there are often special low rates for calls/texts within the same network. And receiving calls and texts is always free when you have a European SIM and you're using it in the country you bought it (eg. with a French SIM and while you're in France, calls and texts are free to receive - from anywhere).

If you need the phone only for emergencies or only to make a few calls, you'd probably do best buying one in your first country, and just putting enough credit on it to last your whole trip. The phone will work in both countries, and the slightly higher cost when out of the country of origin won't be high enough to matter. If you're making a lot of calls or need a local number, you will want to buy a SIM for each country.

I know - clear as mud. Let us know how you intend to use the phone, so we can tell you more useful info.

Harold, thanks for the full course on phones.) My needs are very simple: I will probably only use it to notify a hotel of my arrival time, or to get my passport back as was the case in Cinque Terre some years ago. I may make no more than 2-3 calls total. I carry a laptop for most communication. I’ll be traveling alone by train in Italy and France. No calls back home, just talk and text without data.
I own a travel phone which I bought from BrightRoam (now iRoam) back in 2009. The new company tells me my 2G phone is not supported. It seems the right thing is to buy a phone upon arrival in Italy and get a SIM card that works in Italy and France.

"It seems the right thing is to buy a phone upon arrival in Italy and get a SIM card that works in Italy and France. "

I agree. This way, you will be certain that it will work there.