Hi Eavigda1 -
Welcome to the forum. You have picked a very nice area to visit in June. Early June might still be snowy, but your dates are likely to be OK (mid June, right?). Lots of wild flowers in the meadows at lower elevations; probably some up where you will be
Here is a start, with a small report from a forum member:
Check this link, too.
www.jungfrau.ch/en/tou rism/destinations/klei ne-scheidegg/hiking-trails/
That hike up to Schilthorn could well be fine if you are accommodated to mile high ( and much higher ) hiking, will have no problem with breathing in the thin air at those altitudes, and have hiking staffs and good boots. Plan to start at 1638 meters ( one mile high) and go up to Schilthorn at 2970 meters ( a bit short of two miles high) on steep mountain trails. That is approximately a one mile climb. Plan to pant a lot, unless you have been practicing at high altitudes. However, at the highest altitudes, the trails might be closed...snow and ice.
The forum moderators ( Arno and Annika) can comment if those high trails will be open for the summer season when you are there. Might till be unsafe at the highest altitudes, or closed.
<<"I would love to explore the area but we don't want to spend half our day on trains/cable cars. We want to be out and about enjoying the beautiful scenery."">
The trains and cable cars offer beautiful scenery. However, there are lots of trails, too.
Check the Bernese Oberland link above, and particularly, the Jungfrau Region link within..
This map can help:
if you combine it with the timetable: