I was in the region in June 2015. Specifically, in Wengen, on June 14th (late afternoon), June 15th, and the morning of June 16th.
At that time, we were able to take the cable car from Wengen up to Männlichen, and the trail from Männlichen to Kleine Scheidegg was passable, with lots of snow around, and had only been open a week or so.
This trail takes less than two hours and is one of the most spectacular (and easy) trails in Switzerland. My grandson was with me and we wanted to walk the trail. The clouds were so thick for all except a few short hours just as we arrived on the 14th that we could see nothing to make the trail walk worthwhile all the while we were there.
Since the clouds often come in layers, the Jungfrau or Schilthorn may well have had wonderful visibility ( or may not) and the cloud deck was high enough to make a walk and bus ride on the floor of the Lauterbrunnen Valley pleasant. Mürren was in thick fog (clouds).
The lift (Wengen stop named Wengiboden) to Männlichen LWM up top) started running for the Summer season on approximately May 30. The date for the lift is set in the published schedule at approx. that time each year. The trail opening varies with Spring temperatures (snow melt) and can be as early as perhaps the 15th, or as late as some days into June.
The schedule for Spring will be available after December 15th; however, transit times for various journeys can be estimated from current dates in the timetable.
The reason for the long story is to point out that you may not find many or any of the "ski" lifts open when you are there. You will find the cograil train that runs from Grindelwald to Kleine Scheidegg (branch up to Jungfraujoch) to Wengen and Lauterbrunnen to be running. Runs all year.
So, your effort to make sense of the many possibilities for transport/rides/trips, may, unfortunately, be simplified by absence of many services.
Here are some things that I believe will be open. I ask Arno and Annika to comment. They know the details much better than I do.
Cograil as noted. Rather scenic ride.
Stechelberg to Schilthorn 4 stage cable car lift, via Gimmelwald and Mürren
Cable car from Lauterbrunnen to Grütschalp, and "train" to Mürren.
Bus on Valley floor - Lauterbrunnen to Stechelberg.
Ways to learn more -
A map -
I use this one.
There are more detailed ones at:
map.geo.admin.ch/?topic=swisstopo&lang=e n&bgLayer=ch.swisstopo .pixelkarte-farbe&layers=ch.swisst opo.zeitreihen,ch.bfs. gebaeude_wohnungs_regi ster,ch.bafu.wrz-wildruhezonen_portal,c h.swisstopo.swisstlm3d -wanderwege,ch.swisstop o.swissalti3d-reliefschattierung&lay ers_visibility=false,f alse,false,true,false& layers_timestamp=18641 231,,,,&X=117695.00&Y= 499625.00&zoom=7
A useful panoramic map for the lay of the land with main destinations:
www.google.com/search?q=Jungfrau+region+pano ramic+map&tbm=isch&img il=F-dB3vv18O1cIM%253A%253B Qz-Bn9Am3H3P-M%253Bhttp%25253A%2525 2F%25252Fwww.mappery.c om%25252FJungfrau-Grindelwald-region-summer-map&source=iu&pf=m&fir =F-dB3vv18O1cIM%253A%252C Qz-Bn9Am3H3P-M%252C_&biw=930&bih=55 6&dpr=1.33&usg=__UO8Io QK0NXJprI7TyTHW3Eelqzg %3D&ved=0CCcQyjdqFQoTC LaNxbCft8gCFQoVPgodcyU Mnw&ei=-KsYVraxIYqq-AHzyrD4CQ#imgrc=F-dB3vv18O1cIM%3A&usg=__ UO8IoQK0NXJprI7TyTHW3E elqzg%3D
And - the timetable. Check your dates for particular services (cable cars, boats, etc.) to see if they are running.
Note - The schedule for Spring will be available after December 15th; however,transit times for various journeys can be estimated from current dates in the timetable.
Arno and Annika (the forum moderators) are the experts on rail passes. With your itinerary, and the time of year, they can likely guide you with more specifics than I can. I personally have a multiyear Half Fare Card, which requires that I have mailing address in Switzerland. I do, although I live in the USA. That works for me. So, I really am not familiar with the many options.