Where and when to rent car

Rome --> Cinque Terre --> Interlaken should all be train. All these places you do not want a car when you are there, and the train service is good.
Interlaken - standard warning that there are better places to stay in the area (see multiple posts), some of which are not accessible by road.
Venice is obviously also no car when you are there.

But, what are your plans for "visiting Germany and Austria." between Interlaken and Venice? Depending on what these plans are is the decision on whether you want a car or not.
Also note that a one-way hire with a drop off in another country (say Switzerland to Italy) has a large surcharge.

Do not rent a car in one country and drop it off in another country unless you are prepared to pay a large drop off fee.

German and Switzerland have the most dense rail systems in Europe, and their trains are fast and cost effective. Those are two countries where I would not rent a car.

Hi Gary. Trains are great, but with 4 people, train fares can add up quickly, and of course having a car gives great flexibility. As others said, best to rent and drop car in same country. Based on your plan, it seems like Italy is the country to rent and drop, so you might consider renting after CT (in La Spezia, or after train to Milan) and then drop it when you get to Venice. Enjoy your trip!

You did not provide any dates . Try using an app like Kayak to price compare different pickup/drop-off dates and locations.

I did a sample test on Kayak with a pickup June 8 at Interlaken and drop-off 10 days later on June 18 at Venice, and Kayak gave an average car rental quote of about USD$100 per day. YMMV. Don't forget optional coverages, fuel and the real killer, daily parking. Plus you may incur traffic violations which can be a nasty surprise after your trip.

To take some of the discussion further, if your only intent is to get from point A to B, take a train. Many people say they want to drive to explore, but really only allow enough time for a direct drive. My preference is to take public transport between areas, then rent a car if I need to explore an area for a day or two where public transport is non-existent or very time consuming.

with 4 people, train fares can add up quickly

Unless you are packing very light, most 4 passenger cars in Europe don't have room for 4 people's luggage. Charges for a bigger car can add up quickly. You did not say the age of your children. For most trains, children under 15 are free.

If you are traveling locally, on, for instance, a Bayern-Ticket, each adult is only 6 euro more (and under 15 are free).

having a car gives great flexibility

Flexibility is important if you don't plan. A well planned trip doesn't need flexibility.

Hi Gary -- when I want to explore the countryside, I always rent a car. There are so places where a lack of a car will keep you from seeing cool places. When I'm just visiting the big cities, a car is a headache and not worth the hassle.

Sounds like you'll have a bit of both, so my recommendation would be:

• Train from Rome to the Cinque Terre. There are lots of reasonably fast trains that go from Roma Termini (central station) to the 5 towns of the Cinque Terre. Cost is about €35-50 euros, depending upon how fast the train is. I use the Trenitalia website to plan out my travel options. Note you will likely have to change trains in La Spezia to get to the 5 towns since the towns use a slower local train for connections. Google Trenitalia to get their website address, since I apparently can't post it here
• Pick up a car after you leave the Cinque Terre. The best site I've found and use for all of my car rentals in Italy is Kemwel. You can rent a car with full cover for the best price I've seen, including 1-way rentals. They issue you a rental voucher that's pre-paid, just bring a copy with you when you pick up your car. On Kemwel, you'll need to specify a pick-up city. Around there, your options will likely be La Spezia, Levanto and Pisa -- all of which you can reach by train. I usually pick-up at La Spezia, just train from your Cinque Terre town when departing to La Spezia, then catch a cab from the front of the station to the rental location (on your rental voucher). Note that you can only pick-up your car on certain days & times. The best times are usually between 9-12 and 3-5 on weekends, and before noon on Saturdays. Google Kemwel to get their website address.
• Drop your car on arrival to Venice. Most rental car companies have a drop point at Trochetto car park. If you didn't take the gas option when you picked up your car (not a great deal), be sure to gas up before you arrive at the car drop point.

Cheers, and happy travels.

Michael