We went to Switzerland last summer and have two daughters ages 11 and 10. We decided to do it slightly differently from Rick's advice due to our own interests. We flew into Zurich and spent 1 night there before taking the train through Luzern (spent lunch and a quick tour in town) to Interlaken. We spent 3 nights in Interlaken and then 3 nights in Lauterbrunnen. (From Lauterbrunnen we did take a day trip to Lake Geneva to visit the Olympic museum. and experience the Golden Pass train ride.) Our last night we spent in Bern before flying out the following day from Zurich airport.
Many folks here wouldn't do the same itinerary since they can sight see all day and then move on to a new location/ hotel. Our family doesn't travel well that way. We do better having a base and doing day trips. I suggest reading the guide book and deciding on the places and/or activities your family would enjoy doing. Then decide how many hotel changes you want to make (which also eats into sightseeing and can be very tiring for kids.) Also I strongly suggest doing the lower hills before going into the Alps. They are truly awe-inspiring and after spending time in Berner Oberland, other hills and smaller mountains (like near Lucern) just seem so dinky (which is a bit unfair for all those other lovely areas.)
For us we loved the following things when in Interlaken: Swiss Open-Air Museum at Ballenberg, Kayak trip on Interlaken lake (which won't really be option for you), and St. Beatus caves. You can easily take a day trip to Luzern and Bern as well. Also loved taking the boat ferries on the lake.
Lauterbrunnen was a nice small town, but we wish we had stayed in Wengen. I suggest trying to find a hotel with an indoor pool. There is the cable car that takes you from Wengen to the start of the Mannlichen-Kleine Scheidegg Hike. Very easy with THE most amazing view of Jungfraujoch!!!!! We also loved go up the other side of the valley to Schilthorn for the rotating restaurant and Bond museum. There are a lot of other gondola rides, hikes and of course the famous Jungfraujoch train. We couldn't go because of health reasons. Personally I don't know what the big deal was with Gimmelwald. Definitely wouldn't stay there with kids, but worth passing through. We enjoyed having a drink at the one and only restaurant in town.
We wouldn't return to Bern as a family, though I really enjoyed visiting the Paul Klee museum.
We were very disappointed in the Golden Pass train route to lake Geneva, especially after being in the Alps. Was ho-hum scenery, and the Olympic museum was not worth the special trip there.
You should be fine with vegetarian food, but there isn't a lot of variety in the smaller towns. Interlaken had more variety of restaurants than when we were in Lauterbrunnen, but not all of Rick's picks were really accurate. Did better using his guide for places to stay though.
Have an amazing time!!!