The Cinque Terre take more time to reach than a glance at a map would suggest; Vernazza is 5-1/2 to 8 hours from Perugia by train, for example. You can take a look at train times on the TrenItalia website. Keep in mind that you must use the Italian version of city and station names.
Use ViaMichelin for driving times, keeping in mind that folks say it tends to be optimistic. It guesses a bit over 4 hours for Perugia-Vernazza, plus over 20 euros in tolls (each way).
I'm not dissing Perugia; I think (without checking ViaMichelin driving estimates) it is well-positioned for visiting a lot of wonderful small cities and towns by car, including Assisi, Gubbio, Spello, Trevi, Spoleto, Orvieto, Todi, Cortona, Montepulciano and Arezzo. However, you are traveling with 3 children, so I'm guessing beach time will be important. VM tells me that Perugia (which is located more or less midway between the Adriatic and Tyrrhenian Seas) is about a 2-hour drive from the Adriatic Coast, and I don't know how far you'd have to go along the coast to find a good (sandy) beach.
Also be aware that Perugia is a city, not a small town. If the house you are contemplating is on the outskirts, it might not be within reasonable walking distance of the historic center.