I am traveling this spring in Switzerland, France and Italy. I want to be able to make just a handful of local calls in each country. I will NOT be using my American cellphone to call the United States when I'm there. (Might not even bring it with me if I don't need it.) What are the simplest and least costly methods of making local calls in these three countries-- with emphasis on France? (At the same time, I don't want to rely on local "phone booths".)
You should buy a SIM card in one or more of the countries. You may find that the first one you buy can be used in other countries, or you may have to buy a new one in each country. Be prepared to spend roughly 20-30 Euro for each SIM card. And make sure you have a phone that actually takes SIM cards and is unlocked. If you're coming from the US it may be easiest to buy a "dumb phone" ($35 or so online) for travelling.
If you want data service too, you may be best served by buying a cheap smartphone when you arrive, because the bands are different in different parts of the world, e.g. a US phone used in Europe will often have poor data speeds, and vice versa.
The good news for you is that roaming charges within EU member states are legally capped at a very low level (19ct./min and 6ct./text), so a French SIM card will work quite well in Italy and vice versa. Furthermore, prepaid SIM cards are extremely cheap, as in 20€, including a simple phone and 5€ of credit.
Unfortunately, Switzerland does not belong to the EU, and the roaming rates in both directions (Swiss SIM in EU, and EU SIM in Switzerland) can be horrifying (such as 1.80 CHF / minute). If you plan to spend more than just a few minutes on the phone in Switzerland, I suggest getting another cheap SIM card there as well. M-Budget Mobile has offers starting around 20 CHF, including 15 CHF of credit.
On the other hand, if using WiFi and VoIP is an option, Swiss federal train stations offer 1 hour of free Wifi per day (per station), though I haven't tried to use VoIP with that, so I can't comment on the connection speed.
The cheapest, but not simplest option is to use public WiFi.
For cheap and simple, get a French or Italian SIM, one with low roaming rates to all of Europe.
The EU roaming rates are pretty low, by law. Switzerland is not part of the EU, and the telecom cartel is quite strong there (the biggest provider for this 8 million people country is valued at 27 billion $ for a reason), so roaming there can be expensive depending on the provider.
For simplicity, many EU providers offer the same roaming rates for all of Europe, including Switzerland. That means the cheapest EU SIMs offer roaming rates in Switzerland that are lower than local rates of the more expensive Swiss SIMs.
I can tell you about Switzerland. I was in Switzerland last year and I bought a SIM card from swissCom. The call rates and data charges were pretty cheap. The SIM card costed me 20CHF and was credited with 30CHF (it's actually 20CHF but due to some offer, I got 30 CHF credited to my account). For more information here's a link to Swisscom.