You got some good answers so far. Zürich is expensive and not a particularly scenic destination. I enjoy it because it is a cosmopolitan city with lots of cultural things, but it is not in the mountains.
As noted, Luzern is a good base. Beautiful surroundings; nice old town.
The Jungfrau is a long way from Zürich or Engelberg, or even Luzern .
In order to not only answer your questions about travel times and schedules, but to help you understand the options, you should use the timetable:
Simple on the surface, it has considerably more detailed information if you dig down in the menus. Please read the introductory page at the link I have given you. Be aware that the timetable computer defaults to the fastest route, so it may be necessary to enter a "via" station to force the route to a more scenic one.
These links may also help you:
Pilatus is very close to Luzern. There is not much to explore up top. Half a day is plenty.
Many visitors enjoy the Rhine Falls, but I prefer to see things that are uniquely Swiss. I have seen a lot of waterfalls, so I don't go to the Rhine Falls.
Please give some thought to using time to stay near the Jungfrau or at Interlaken, instead of seeing the Rhine Falls and staying at Engelberg.
The larger region is called the Bernese Oberland, with the Jungfrau region a special part of it.
The Jungfraujoch visit is quite expensive, and should not be done if there are clouds at high altitude. No views, then. But the entire area near the Jungfrau is beautiful, and there are less expensive options, such as Schilthorn, or the villages.
By the last week in June, the Summer season will have begun, and almost all hotels, restaurants, tourist services will be open. There may still be snow on some of the higher trails, if you plan to hike.
This link will give you some information about the Glacier Express route, as well as other scenic routes:
www.sbb.ch/content/dam /sbb/en/pdf/en_bahnhof -services/en_am-bahnhof/en_unsere-bahnhoefe/panoramic-routes.pdf