1st Time travel to Zurich, Munich & Salzburg

With all due respect before you plan dates I'd read guide books to determine what you want to do then decide how long you want to stay in each area. My personal opinion Salzburg has a lot to see and experience both in Salzburg and around Salzburg. Same could be said for Munich and Lucerne more so than Innsbruck or Zurich. I could spend the whole week in each of these places and find much to see and do.

This could be done easily by train. Rent a car in Zurich and it will come with a Swiss vignette already. You will need to get one for Austria at the border. Driving to Munich, you will pass through Austria, so get it there. It is a sticker that goes in a certain spot on the windshield and can be thought of as a toll sticker for Austrian Motorways. Get caught without one and its a big fine.

If it was me, I'd fly to Munich from the start and go by train. Munich is not a very car friendly town. None of the cities are, but Munich especially. Assume you will be paying for parking everywhere on your wish list. Without a car, you could skip Zurich altogether. There are direct trains from Luzern to Zurich Airport.

I agree with Sam. Very easily done by train, and if you haven't bought flights then fly into Munich and out of Zurich. I believe you are trying to see to much in too little time. If you really want to see Munich, Salzburg and Lucerne then I would spend 2 nights in Munich, 2 nights in Salzburg and 2 nights in Lucerne. If you are traveling by train getting to the Lucerne airport on the last morning before you departure should be an easy 1 hour train ride. Skip Innsbruck, Bern and Zurich.

In addition to the above (ie train) if you haven't purchased air travel, rather than just spending a night in Zurich then driving to Munich, why not fly into Munich and depart from Zurich.

A Vignette is a sticker you buy to pay Tolls for using Autobahns. In Switzerland and Austria you pay tolls not per Km, but per week (or month or year). You buy the Vignette before going on an Autobahn, and attach it to your windscreen to prove you have paid. Any car rented in Switzerland will already have a Swiss Vignette for 2018, but you will have to buy an Austrian one at the border.

July 18 - Travel by car to Munich via visiting Neuschwanstein castle & other Ludwig's castles - Total travel time approx 4 hrs. Stay at Munich.

Where are you getting these timings from? According to https://www.viamichelin.com/ Zürich to Füssen (location of the castles) is 03h01 238 km and Füssen to Munich is 01h58 133 km. These figures assume you stop for nothing and now exactly where you are going. You need to add 25%, plus a lunch stop. I do not think you can do this and see the castles in one day.

You have only one night in Munich, arriving late and departing early for Salzburg. You will not see anything, you would be better off going direct to Salzburg.
You also have 1 night Salzburg and 1 night Innsbruck. Once you allow for travelling you will have no time to see these cities. "One-night-stands" are not a good idea. Better to stay at least 2 nights in fewer places.
This is a 7-night trip. I would restrict it to two locations, or a maximum of 3 (2+3+2 nights).

All your other driving time estimates are too short. Salzburg - Innsbruck, you say 1h20, Viamichelin says 02h10 183 km (+25%), and that doesn't allow time to find your hotel, park, check-in and then proceed on foot to see the city centre.
Innsbruck - Luzern - Bern, you say 3½hrs, Viamichelin says 03h45 304 km + 01h19 112 km.

Note all the cities you list (Zürich, Munich, Salzburg, Innsbruck, Luzern, Bern) have city centres where no cars are allowed. You cannot use a car to drive round the places you want to visit in these cities. And most city centre hotels do not have parking. You need to check on parking before you book a hotel.

On your last night, I would stay in Luzern instead of Zürich, and start out early for the airport.
When in Luzern, you can wander around the old town centre, go on a boat trip on the lake, or take a trip up one of the local mountains (Rigi, Pilatus or Titlis) by train or cable car.

All the locations you have listed are connected by train, it would be a lot less stress, and would be compatible with the car-free cities, to do this by train (with fewer stops).

I have visited Munich three times. It is a wonderful city with so much to see. It is an easy walk from the train station down into the city. The mechanical clock tower is a great experience. The park is beautiful. The often have lovely concerts there and the Rose gardens are beautiful. At one part of the river in the park, they actually surfboard across the fast moving River. I would research this lovely small city. I took a train to two of Ludwick's Castles from Munich. It took almost a day to see each one. I also agree with Salzburg. A lot to see. With such a short amount off vacation time...I would leave out some off your choices. You can see some lovely towns by train from Munich.

Things to do in Salzburg:

Sound of Music Tour

Visit a salt mine via tour or do it yourself by asking at train station for a "Salt ticket" (train plus bus to one)

Day trip to Hitler's Hideout/Eagles Nest (Google image it)

Tour or train to Werfen for visiting the ice cave (Google image "Eisreisenwelt")

Day trip to Hallstatt (Google image)

Thank you everyone for taking the time to give your expert advise.. You folks are great...!These are very valuable and helpful tips. I was looking at train travel and it seems more expensive for 3 people, more time and no freedom of travel. It's cheaper to drive..! Maybe I'm wrong... :) Perhaps we can stay based in Fussen and travel to Austria and back to Lucerne. Any other tips are most certainly welcome...!

You have a week of basically one-night stays. This is not The Great North American Road Trip; This is a European Nightmare on Wheels.

First, you need a good guidebook, like Rick Steves' Switzerland. Then you'll have guidance on passes, where to stay, what to see.

Cut your locations down. If you have already purchased airline tickets, cut out Munich. It is an outlier and far too much travel time. Consider using trains and stay in Switzerland for the week. OR once you land in Zurich, fly to Austria if you must see Salzburg and Innsbruck, but that leaves little time for Switzerland. You could do 2 nights each in Salzburg and Innsbruck then 3 in Luzern. It is a 4.5-hour train trip between the two You can get to Zurich for your departing flight in time from Luzern as it is only about an hour by train, Do not waste a precious night in Zurich.