Side trip to Florence

What do you mean by "mostly" in France? Will you be in other countries? A lot depends on where you would be coming from.

Regardless, with 3 weeks at your disposal, you certainly could spend 4-5 nights in Florence and Rome. The easiest way to look for multi-city (open jaw) flights is to use Google Flights - https://www.google.com/travel/flights. Make sure you choose "multi-city" when searching for your arrival and destination, instead of the default "round-trip." You will find many low-cost options like Easy Jet and Wizz for a total of around $100 or so, or even less depending on when you are flying.

Since you are both art lovers, you should definitely spend 4 to 5 nights in Florence just for the art and architechture, and not go to Rome at all. If you get an inexpensive flight from France to Florence you'll not lose as much time as on a train. If you try to squeeze in a trip to Rome, you may find yourselves a bit rushed. If you do not spend most of your time in France, you might consider spending a week there and a week each in Florence and Rome. Just depends of what hold the most interest for you.

"It seems like they want to charge double"

Who is they, and double compared to what, and from where to where? It's hard to make any recommendations with no idea of where you are starting from, where you are visiting in France or elsewhere, and how many days you could devote to an Italian "side trip".

If you can spare 4 days, then flying to Florence could certainly be worthwhile if you are art lovers. Look at Skyscanner for low cost one way fares. If you have 6-7 days, then you could fly from France to Florence, then train to Rome.

Assuming you would be flying into Paris from the US, you could certainly look at a multicity ticket into Paris and out of either Florence or Rome. Use Google Flights to look at your options. Book tickets directly with the airline.

We have been flying open jaw for 40 years and it is generally the average of the two round trip flights. They don't 'charge you double.' You book multi city (not one way flights). We almost always do that. Flew into Paris and out of Berlin this spring for the same price we are flying round trip to Paris this fall. We have flown into Paris and out of Madrid. Into Vienna and out of Amsterdam etc etc. If you are spending 3 weeks in Europe it would be easy to fly into Paris and then out of Rome and visit Rome and Florence.

To get between Paris and Florence or Rome, your cheapest bet is likely to be a budget airline like Vueling or Easyjet.

CJean, I have been looking to use my credit card miles on Capital One. They don't have a box to check for multi-city.

I use Capitol One credits and book on the airlines own sites multi city or round trip whichever suits and then get reimbursed by Capitol One.