After Breakfast you hop on train in Rome and at 200 KPH you are in Florence for lunch. The train from Barcelona to Madrid is a breeze. Get on one at London Kings Cross (see Harry Potter) dine on the train and in come to Edinburgh. Paris to Brussels or Amsterdam does not take long.
I love to drive myself. I have driven on the left side of the road in three countries. I have driven in twelve countries (not counting home). Let me say that speeding through the Alps is fun, but you don't see much. It is eyes on the road. Both the train and the car give you different sights. But, driving is not as good as riding the train for the driver. I have to listen for my passenger to tell me what they saw.
The benefit of driving is it is slower and allows you to stop when you like. You will need an International Driver's License which is easy to get at any AAA office,
The photo above was taken on a train though Northern Italy. It is a sunflower field. You could see it both ways, train or car, whichever you prefer.