I don't think anyone can suggest an itinerary without knowing what you want to see/do in those countries.
In any case, 10 days is hardly enough time to visit 3 separate countries.
If you intend to fly between the countries, you will effectively lose 1/2 to 1 full day each time you move on.
Have a think about what you want to accomplish and post back when you have a better idea of your travel plans.
So that would be 3 days in each country, minus travelling time?
You could, I suppose, fly into Paris, fly to Barcelona and fly to Rome and then home from Rome. The first time we went overseas we diverted from England for 3 days Paris, 3 days Rome 3 days Athens and it gave us a lovely overview of those cities, all of which we have since visited again.
But it would be very easy to spend all 10 days in just one city with maybe some short trips to nearby towns by train.