depending on your age/s you may qualify for a student rail pass which would enable you to be as spontaneous as you want:>

yes if you are under 26 you can buy the various Youthpasses, which if traveling more than a few times on trains can pay for itself - in most countries still can hop on and off any train anytime.

UK- London- Paris- Amsterdam- Belgium- Rome. All by train. No need for a car.