Airlines for travel within Europe

It's a short flight. To me it doesn't matter which one. Just Google "Rome to Paris flights" and you'll see the options.

Air France
Alitalia
Vueling and Easyjet into Orly--both budget carriers.
Joon

just search for the direct flights

My preferred budget airline in Europe is easyJet, and that would be a good choice from Rome / FCO to Paris / ORY. Be sure to read the Terms & Conditions for whichever airline you choose, especially the luggage rules. If you decide to use easyJet, I'd suggest paying for Speedy Boarding and advance seat selection, as it makes for a much more pleasant budget airline experience.

Note that the cheapest tickets are non-refundable and non-changeable once purchased, so choose carefully.

go to the wiki page for the Rome airport. There is a chart showing all the airlines and destinations. Use the "find" function (ctrl+F) and type in "Paris". Note which airports in Paris - sometimes Beauvais comes up - not exactly near Paris.

For this type of short inter-Europe flight even the legacy carriers like Air France will have unfriendly luggage rules so be sure to check those carefully.
Kayak is an easy way to research your options on the flight ; once you see what is available I would likely go to the carrier's website to read about the luggage and make any bookings directly with them rather than through Kayak or similar site.

I would go with whichever direct flight costs less or whoever has the best times for me still within budget.

Probably best for you if you fly into ORY instead of CDG ; smaller which is usually better for getting out upon arrival.
If taking a taxi ORY will be cheaper and faster to reach the city but I believe if you plan on taking Public Transit from the airport to the city than CDG is better.

Check out Skyscanner.it. It will give you a variety of options. I've used Ryan Air a couple of times, which was fine and recently booked a trip on EasyJet. I like the way EasyJet posts their prices - easy to see and calculate - no guessing as with some of the other airlines like Wizz, who seems nearly impossible to contact.