Matt,
I've travelled that route several times and it's easily done by train. However it will be a full travel day.
The best option is the one that Lola mentioned above. While the 08:00 departure from Interlaken Ost has four changes, it's the shortest trip at 7H:23M and will get you to the Cinque Terre by 15:23, just in time for a brief walkabout and a glass of Prosecco in the small harbour, and then a fine meal.
Using a car on that route is not a good idea. In addition to the hefty drop fee, there's no way you'd be able to drop the car in Vernazza as it doesn't have a rental agency. You'd probably have to go to La Spezia and then backtrack to Vernazza. The would be a darn nuisance! You'd also need an IDP for driving in Italy and would have to be vigilant to avoid ZTL (limited traffic) areas in any towns you pass through in Italy (again, hefty fines).
If you use the 08:00 departure as mentioned above, the train from Milano Centrale to Levanto is an Intercity, so there will be compulsory seat reservations, which are specific to train, date and departure time. Reservations will be provided with your ticket. If you were planning to buy advance tickets, I'm not sure you'll get any discounts on the Swiss rail trips, so you could easily buy your tickets for departure when you arrive in Lauterbrunnen. You could save money with advance tickets for the trip on the Intercity from Milano Centrale to Levanto. The trip from Levanto to Vernazza is via Regionale, so there will be no discounts.