Have the agent validate the pass before your first use, at the Zurich airport station. They usually will want to check your passports and see the kids and make sure you are the people on the pass.
If you are going from the airport to Lauterburnnen that day, it will cover the whole ride. That will start the 30-day period for using the pass. The pass will have 3 little boxes where you enter the date for the day you choose to use it for full coverage. Be sure you enter the date before you board the train, or the conductor will scold you--or possibly levy a fine. For the other (non-covered) days, just show your pass when you buy a ticket and you'll get the 50% discount.
On Day 5, when you travel into Italy, the pass will actually cover to the first stop beyond the border, which is Domodossola on the courte you will be taking. I would buy my tickets to Vicenza at the Lauterbrunnen station before starting this journey. Otherwise you would have to do it when you change trains at Spiez or Brig. Again, show your pass and they shoudl price the tickets just onward from Domodossola. Make sure that the route you are taking goes to Milano Centrale, not one of the other Milan stations.
You can see the coverage for the Swiss pass (or Flex pass) if you Google
"Swiss synoptic map"---it is the same map that comes with your pass. It will show all the trains, boats, lifts, etc. and which ones are only covered 50%. And you can see the schedule and routes for each of your journeys at www.rail.ch
Do not worry about asking too many questions. It's confusing if you havent done it before. And have a great time--we'll be a few weeks behind you, landing in Zürich and heading for Italy.