Hi jahash -
Welcome to the Forum.
That sounds like quite a nice trip.
To make sure that you have sufficient time to do more than travel, take a look at the official Swiss timetable and check the time of your journeys.
The first day seems very long. You might give thought to whether you will stay in Luzern or Interlaken.
If you take the TRAIN all the way from Milano Centrale to Interlaken, via Luzern, you have a 6 and 1/2 hour journey. I could not use a real starting point, because you did not have one decided yet. Clearly, it will take some time to get to Milano Centrale, if that is the route you choose. Add that time.
If you add in the boat trip - Flüelen to Luzern - for the Wilhelm Tell route, that can add something like an additional 1 hr 50 minutes ( approximately) for boat vs. train.
Although the boat part of the Wilhelm Tell is spectacular, especially at the southern end, if (and only if) you are pressed for time, you could skip the boat and take the train directly to Luzern. May have to change at Arth-Goldau. Easy connection.
In what part of June are you traveling?
The end of the low season in the high alpine regions comes in late May. The best skiing is past, and the summer hiking is not fully open.On the trails above 2000 meters, the snow may still block the way.
June is Spring at lower altitudes, but not really completely finished with Winter at high altitude until about mid-June; sometimes later. The weather pattern has not yet settled into summer, so there may be quite a few rainy or cloudy days, mixed with the sunny ones. The wildflowers will be blooming in the alpine meadows.
The cograil to Jungfraujoch will be running.
Not all hotels will be open until later in June, although there will certainly be rooms available.
When visiting the Berner Oberland - particularly the Jungrau region - it is wise to plan on a couple of days in the area, or have a flexible schedule to adapt if clouds at the altltude of the Jungfrau obscure the (expensive) views.
And, keep an eye on your spelling. The time table is forgiving, but you will have an easier time if you spell the names correctly.