Help planning time in Switzerland!!

I'll leave most of your questions to people who have been to Switzerland--I'm going for the first time this summer. But regarding trains, I can help you. Here's the website for SBB, which is the best resource for Swiss rail schedules: https://www.sbb.ch/en/home.html

From Zurich airport ("Zurich Flughafen" on the site) to Mürren involves 4-5 changes; to Wengen, 3 changes. To reach either village, you first go to Interlaken Ost station, which requires one change in Bern; there's also a scenic Luzern-Interlaken express if you changed trains in Luzern instead, but that adds time to your travel so you'd have to weigh those options.

Once you get to Interlaken Ost, you take another train to Lauterbrunnen. From there the routes to Wengen and Mürren diverge ... and also get increasingly scenic, as this is where you start heading uphill via cogwheel trains, funiculars, and the like, depending on which village you're visiting.

Wengen is more convenient to Jungfraujoch; Mürren is more convenient to the Schilthorn. Regarding Harder Kulm, it's on the other side of Interlaken from both Wengen and Mürren, so your choice wouldn't matter for that destination. If I understand correctly what I've read, Trummelbach Falls is in the Lauterbrunnen Valley, so I think it would be accessible from either village, but others who've been there can correct me if necessary.

Based on my experience booking a room two months ago (for early July), I'd choose your village and get your accommodations as soon as possible.

How many connections do you make on the train from Zurich to Murren or
Wengen?

Zurich > Interlaken Ost > Lauterbrunnen > Murren or Wengen. Everything is well marked, so it's impossible to get lost.

We want to do Jungfraujoch, Trummelbach Falls and possibly Harder
Kulm. Are these two towns easily accessible to these sights?

Yes, very much so.

If you stay in Murren, consider letting your kids hang out at the Mountain Hostel down the hill in Gimmelwald. It will have lots of kids just a few years older than yours. Pizza, games, music - lots of fun.

May also want to do an EASY hike somewhere, so suggestions for that
would be great.

There are tons of easy hikes to do in the region. The train station/TI in Lauterbrunnen will have maps, and your hotel will probably have them too.

Is Interlaken worth a visit?

Not really. If you get lousy weather it may be worth a couple hours to expore. It's mostly a transportation hub - but the lake cruises depart from there.

Wow this is awesome info! A few more questions if you don't mind me asking. We want to go to Rome after Switzerland. Is it a doable train ride or should we fly? I would rather take a train since we will definitely fly somewhere else in Europe from Rome, possibly to Germany.

sepalma, to get to Rome, you can look at the Swiss rail site sbb.ch/en, train line.eu, bahn.com, or whatever site you feel comfortable with. What does "doable" mean to you? Many options for trains: looks like anywhere from 8-11 hours, with 3-5 changes. If you compare to flights, you have to take into account how many hours it will take to get to an airport, wait after check-in, fly, and then get from the airport to the city in Rome. So don't be fooled into thinking it will automatically save a lot of time to fly, even if that is simpler.

Dear first timer,
The first thing you want to do is nail down your flights and accommodations, ASAP. You said "may fly into Zurich" so I'm guessing you don't have that yet. Zurich is good, but if that doesn't work out, Geneva will work as well. Both are about 3 hrs away from the Berner Oberland.
As for Murren vs Wengen, with teens I would definitely recommend Murren. Chances are they will have more energy than you, and so they can take to trails if you want a nap. In Wengen, not much trailwise, but Murren has 5 good ones (to Spielbodenalp, to Grutschalp, to Gimmelwald, and left and right off of Allmendhubel). You can't get lost. I've got some pics and maps I can send you. Click on your name in the upper right of this page to check your mail.)

Also, in Murren you all could go up to the Schilthorn as a mountain peak. It is cheaper than going to the Jungfraujoch, and you can do it in a couple of hours from Murren as opposed to pretty much all day going up and back from Jungfraujoch.

If you can't find accommodations in Murren then I'd recommend Lauterbrunnen, Grindelwald, Wengen in that order.
Harder Kulm is nice, but before that I would recommend the Mannlichen to Kleine Scheidegg trail which takes about 2 hours and is almost flat. The views from there are even better than Harder Kulm and the trail offers many vistas while Harder Kulm only offers one. If you can take only one trail in the Berner Oberland, Mannlichen to Kleine Scheidegg is it. Save it for a clear day, while Trummelbach could even be done in the rain.

Clarifying what FastEddie said about pizza at Gimmelwald's Mountain Hostel--According to the Rick Steves website update, the restaurant recently closed at Mountain Hostel, and when I pulled up the Mountain Hostel website, it didn't mention a restaurant anymore, other than a breakfast buffet.

The Mountain Hostel never had much of a restaurant. They offered pizza and small selection of other items, but a wealth of beer. They still served pizza as of June 2017.