Going on the Train?

My girlfriends parents are coming to Europe in July with the ambition to see some of our older cities. As it’s super expensive to get rid of a car in central London, Paris and Rome I think we will use the train to get around. I definitely prefer driving, but giving the locations I think the train would be just fine. There is actually one trip I’m even looking forward to. The ride from London to Paris is supposed to be pretty quick. And so are some of the other French ones? They are suppose to have a few that travels in the area of 300 kilometers per hour I think?