To rent or not to rent when crossing borders

A huge answer to that question is the financial one of whether there are dropoff fees between your rental origin and destination. It can vary by country and rental company, and even exact cities. I have found some combinations had the large dropoff fees while others, between the same countries but different cities, did not.

And what experience do you enjoy: the freedom of a car or the ease of being transported by a bus or train?

I like to take trains if I'm only traveling between large, well connected cities.

2 week tour with friends

A key piece of information is the number of friends. 1 or 2, I'd go with trains. A car will be a pain in cities like Lyon, Turin, Milan. Dijon to Lyon is connected by TER trains, rather inexpensive and price is always the same price and no need to book in advance. Cities on the way to Lyon are Beaune, Chalon sur Saone, and Macon. all wine towns. Lyon to Turin to Milan to Switzerland all with frequent trains.

If you want a car, two options. Rent in Dijon and use it to work your way to Lyon and drop it there, then train to Turin, or if you have a stop in France planned, say Aix les Bains, or Grenoble, drop it there.

The other option, have your friend drop you back in Switzerland, probably Basel and rent a car in Swtzerland. It will then be returned there at the end of your trip, avoiding international drop fees, and it will come with a Swiss vignette, so you can drive on motorways in Switzerland.

What about crossing borders between countries? Is there a fee simply for crossing? We plan to be in Switzerland, cross into France, then France into Italy, and then back up through the Swiss Alps. Anything we need to pay for to drive in these places that we need to know in advance?

Kim, there is (usually) no issue crossing borders. You should inform the rental company just in case there are any issues or any extra paperwork is needed.
The issue is returning a car in a different country. A French car returned in Zürich has to be driven back to France, by a paid employee, who then has to get back home. Hence the large charge.
Between France and Italy you will not even be stopped at the border, just a road sign. Switzerland is not in the EU, so there is customs (goods checks) but still no passport control. They are mainly interested in commercial vehicles (carrying goods), not in car travellers.

Other issues
- If your driving licence is from a non-EU country, you need to get an IDP (International Driving Permit) in order to certify your licence as valid. You need to get this before you leave home.
- France and Italy have tolls for autoroutes/autostrada, you pay by distance. For Swiss Autobahn/autoroutes/autostrada you pay a flat toll, you need to buy a toll Vignette at the border (CHF 40).
- Make sure you learn road signs and traffic laws for each country. Too many tourists don't and then complain when they get tickets or worse.

The biggest issue for crossing borders is the drop charge. I price different options before deciding. In your case options include trading the car for a train when possible to keep from cross-border drop charges. The benefit is relative. I paid a (relatively small) drop charge for picking up a car in Toulouse and dropping in Bordeaux last June. You can also drop in the same country, train across the border, then pick up another car - again price it out to see if that's a good option. You could also rent in Zurich take your friend to Burgundy and she could train back, then drop the car in Zurich at the end.

A lesser issue can be local regulations (like a vignette for driving on highways) but those aren't usually too bad. Generally you can pick up whatever you need at the border - look for something like a truck stop. In Austria it would be easy to get a ticket, if you didn't stop for a vignette. The police wait at the border, just after the truck stop, and pull over cars without the sticker. Still, it's a ticket - not the end of the world. Often a rental car will have what you need (good reason to explain your plans when booking, and picking up, the car).

Switzerland is not in the EU, so there is customs (goods checks) but still no passport control. They are mainly interested in commercial vehicles (carrying goods), not in car travellers.

Chris, My experience entering and leaving Switzerland in April was interesting, although maybe the situation was normal. We crossed into Switzerland from France at Vallorbe, and the official at the border was just waving all cars through, no need to even slow down; I did slow down just enough to say "Vignette?," although I had an idea where to get one. Then 4 days later, on our way back to France through the same border point, we were again waved through, but on the other side, where we had entered, there was a long line of stopped traffic. Maybe it's random, or maybe there was a reason.

Robert, yes that's typical. Usually the customs just waves everybody through, and only stops somebody who looks suspicious, likely to be carrying goods above the limit. Typical cases reported in the local paper are one-man businesses (shops, restaurants) buying wholesale over the border and driving back in their private car and hoping the customs will not notice. Or people buying wine in bulk from vineyards (the limit is 5 bottles). Of course I have never done this.
Customs posts are only on main roads, and often closed at night. On minor roads all the old customs posts are closed, they rely on mobile patrols.

This news story is about someone who was caught with 120 Kg of meat in their car, and another case with 335 Kg in a delivery van: https://www.schweizerbauer.ch/vermischtes/allerlei/fleischschmuggler-erwischt-49999.html

As for the Vignette, on the autobahn crossing near me they have two lanes, one for vehicles which already have a vignette, another where you stop and buy one.
Photo: https://mapio.net/images-p/25888767.jpg
On non-Autobahn, you need to park up and go into the customs office. Or if it is unmanned, find a fuel station or post office which has one, before you get on an Autobahn.

This can be an issue in the US too. Someone I knew's office was once involved in prosecuting a case where a man drove from Tennessee to Arkansas to buy wine for his daughter's wedding where it was cheaper and lower taxes were paid. Last I heard he ended up having his car (a mercedes) confiscated for this illegal activity designed to avoid local higher alcohol taxes.

A few years ago, we rented in Avignon for a 2-week visit to Provence. We thought we might go into Italy, so I asked if that was allowed. We were told NO by the rental agent. However, when I checked later it turned out the issue was insurance -- if you take the insurance from the rental company it may only be good in France. If you use insurance from your credit card company, it works everywhere.

I think Italy is a special case. When I used to use credit card insurance, I believe Italy may have been excluded