Interlaken means "between lakes", and the town sits on exactly that: the flat delta called the Bödeli, where the Aare runs from Lake Brienz into Lake Thun. An Augustinian monastery was first mentioned here in 1133 and closed in 1528, when Bern turned Protestant. The visitors came around 1800, after painters discovered the view and the Unspunnen festival of 1805 drew a crowd, and the Kursaal of 1859 and a railway up to Lauterbrunnen and Grindelwald in 1890 did the rest. The town itself is a gateway, not the reason you come, and it's honest about that. The reason is behind it: waterfalls, a railway through the Eiger, and two lakes with boats on them.
This is the warm-season plan. It works from late May to September for everything, and with some closures in April, early May and October. If you want snow and skis, the winter plan for the same mountains is 5 Days in Grindelwald and Wengen. Five days is the right length because the mountain days depend on the weather, and you want slack. Interlaken is the base because every sight is on a train, bus or boat from here, in 15 to 45 minutes. The catch is price: this is one of the most expensive corners of Switzerland.
The plan goes day by day: Interlaken and the Harder Kulm sunset, the Schilthorn and the Lauterbrunnen valley, the Jungfraujoch and the Eiger Trail, the Schynige Platte to First hike, and a day on Lake Brienz with Giessbach and Ballenberg. Start on a Monday. The Jungfraujoch lands on Wednesday, with Thursday and Friday as spares, and day five, the lake day, still works in bad weather, so swap it in whenever the mountains are in cloud. Going on to Lucerne? The direct Luzern-Interlaken Express takes under two hours, and 5 Days in Lucerne picks up from there.