Take the 10:45 direct train from Zurich Flughafen (airport) to Interlaken Ost and then switch trains there to head to Murren (Interlaken Ost>Lauterbrunnen>Grutschalp>Murren)
I wouldn’t recommend walking around Interlaken. You’re going to have your luggage to deal with and it would be much nicer to just get up to Murren and then spend your time walking around there. You’d even have time to go up to Allmendhubel for lovely views after you check in or a light hike.
In my opinion Schilthorn is non negotiable if you’re staying in Murren. It would be like going to Paris and not going to the Eiffel Tower :)
Weather dependent I’d do something like this…
Day 1 - arrive Murren around 2 pm (10:45 train from the airport), check in and go up to the Allmendhubel and do the easy walk down to Murren via the Bluemental Trail, Wander the village and Dinner
Day 2 - Schilthorn & Birg in the morning, afternoon walk down to Gimmelwald (the wood fired pizza at the mountain hostel is great) - you could also do the Northface trail in the afternoon and then end up in Gimmelwald for late afternoon/dinner (to do this, ride up the Allmendhubel and hike counterclockwise ending at Schilthornbahn cable car which will take you down to Gimmelwald)
Day 3 - Murren > Grutschalp > Lauterbrunnen > Wengen > Mannlichen > walk to Kleine Scheidegg > option to ride up to Eigergletscher / Eifer Trail or continue walking to Wengernalp from KS and then train back to Wengen > Dinner in Wengen and then return to Murren
Day 4 - Murren > Grutschalp > Lauterbrunnen > walk or e-bike the valley > Afternoon you could do Schynige Platte or a boat ride on Lake Brienz or Lake Thun
Day 5 - Train to Zermatt, Village Walk
Day 6 - Gornergrat
Day 7 - to Italy