Rome to Paris

Best and cheapest are not always the same. Probably the cheapest would be an advance fare purchase on one of the discount airlines. You don't have a lot of choices - train or plane is about it. Driving is out of the question.

Try Easy Jet or Vueling air. Vueling does not have the baggage restrictions Easy Jet does. Flight time around 2H price E59+

Fly, I flew Vueling( but have flown Easyjet too and think both airlines are good) . Its 1.5 hours. It can be cheap if you book well in advance.

Andrew, As the others have mentioned, using a budget airline on that route is usually the cheapest and quickest method. It will be longer and more expensive to go by train, although that's also possible. Both EasyJet and Vueling travel from Rome / FCO to Paris / ORY, so the transportation to and from the airports will be exactly the same in either case. Note that budget airlines often use a "sliding scale" so prices will increase as the flights fill. The cheapest tickets are often non-changeable and non-refundable. Be sure to read their Terms & Conditions carefully, especially related to luggage sizes and weights. If you'll be checking luggage and it will be over the weight limit, it will be cheaper to pay for the excess weight at the time of booking rather than at the gate. One final (but important) point to note is that some airlines only allow ONE carry-on item per passenger (and they do enforce that). Happy travels!

In this case I would say beat and cheapest are the same: fly.