Traveling from France to Italy

Plane or train?

I cannot help with the former ( once we arrive in Europe we like to keep our feet on the ground) but I did check the trains. It is a journey of 8 hours or more from Lyon to Venice, but you could break it into 2 segments of roughly 4 hours each with an overnight stop in Torino. That is my sister's favorite city in Italy ( and she has spent a lot of time in Italy).

We are planning our route in this order: Paris, Normandy, Reims, Burgundy (Beaune) then over to Venice, Cinque Terre, Florence and then end in Rome

All this in 3 weeks? Wow

This will give you an overview of options. Drill down to get to specific vendors.

https://www.rome2rio.com/map/Beaune/Venice

Looks like whatever the choice, most of a day is spent traveling

Which method of transport are you using? Car, train, other?

You can look up driving times on the viaMichelin site: https://www.viamichelin.com/

Time: 08h40 (assuming no stops, add 10-25%)
Distance: 813 km (749 km on motorways)
Costs: €204.89 (Tolls €130.80 + Fuel consumption €74.09)

You can look up trains (but not buy tickets for this route) on the DB website: https://www.bahn.com/en/view/index.shtml
Example for a random date in April (there are later trains as well):

Beaune(Chagny) dep 06:00, Lyon Part Dieu arr 07:44
Lyon Part Dieu dep 08:31, Milano Porta Garibaldi arr 13:50
Milano Centrale dep 14:45, Venezia Santa Lucia arr 17:12

You can also fly from Lyon to Venice, but you need to get from Beaune to Lyon airport first.

I would say this is too far for one day. You need to overnight somewhere. perhaps Switzerland, or Milan?

Ambitious itinerary.
First, you could make Reims a day trip from Paris. Cut down on hassle.
Second, you could drive back to Orly from Beaune and fly to Venice. Easy Jet has direct flights from 34€. Anyway, that's what I'd do.
A flight from Lyon is going to have you changing at Orly or CDG anyway.

A flight from Lyon is going to have you changing at Orly or CDG anyway.

Nope, both Air France and EasyJet fly from Lyon to Venice. See here: https://en.wikipedia.org/wiki/Lyon%E2%80%93Saint-Exup%C3%A9ry_Airport#Airlines_and_destinations

Eight locations in 3 weeks is a very rushed schedule, particularly with 2 big hitters in Paris and Rome. Paris is rushed in 5 full days, particularly as you will lose at least the first day to jet lag, so for a first visit, you could easily spend a week here. Rome is also a 5 day city and Venice 3 full days, so that doesn’t leave much time for all your other destinations. You are going to lose at least half a day every time you move destination.

Flights for May have been available for months, so book asap for the best prices. I like EasyJet, as they are reliable and have no weight limit on hand luggage.

Then it's a no-brainer: Lyon via Easy Jet.

We took the flight from LYS to VCE a few years ago. We flew over some beautiful mountain scenery in the Isère north of Briançon and on into Italy north of Torino so I would recommend trying to get a window seat for the flight, especially if visibility is good.

we have been doing three week trips for the last few years and I feel a minimum of 6 nights per stop allows a comfortable trip. otherwise you spend too much time traveling and not enjoying. I also like to limit travel time between destinations to 3-4 hours for a little sanity. our previous trips below. I always bite off more than I can chew... last year was too many stops and a lot of train travel from France to Italy
Paris is Fabulous spend a week there

1. Paris-Colmar-Lugano-Lake Como-Milan-Bologna-Padua we took trains between locations 26 nights - too many stops.

2. Madrid-Seville-Barcelona train between locations 3 weeks perfect time

3. Athens-Naxos-Santorini-Mykonos-Athens Flight to Naples Italy Sorrento-Positano-Rome by train great trip 3 weeks

this year

Paris-Vienna-Prague 3 weeks

You will find many short-hop flights at www.skyscanner.com

I would certainly fly from Lyon to Italy, both for price (if bought well in advance) and time.

I you are not experienced with these "budget" flights, be sure to read all the instructions on the carrier's website because the rules can be strictly enforced. Also buy the inevitable "extras" ahead of time.

I would add, if you use the sites like skyscanner to find the flight, I usually then go to the carrier's site to book it, especially if you are checking a bag.
I agree this itinerary for 3 weeks is waaaay ambitious,

Venice, Cinque Terre, Florence and then end in Rome (departing) has you criss-crossing the country. Is this by train? Last year I took the train from Cinque Terre to Florence was almost 4 hours. The more segments you rely on the trains, the more time you spend just on the traveling. You could easily spend 3 weeks just on your French itinerary, or 3 weeks on just your Italian itinerary and have a great adventure.

Sounds like a wonderful trip! We are flying on EasyJet from Lyon to Italy. Book in advance— usually only one flight a day.