Best day trips from Zurich?

I haven't been to Switzerland yet, but am going in September 2014, so I've been researching a lot.

Unless you have free accommodations in Zurich, the consensus is that Luzern (only an hour by train) is much nicer and makes a better base. From Luzern, there are lake cruises and easy excursions to Mt. Rigi and Mt. Pilatus, if you tire of the town itself.

The other point on which I see unanimous opinion is that Interlaken is near but not in the Alps, and so time in that region (the Berner Oberland) is better spent closer to the scenery. You have the choice of Lauterbrunnen (in the valley, with access to mountains on both sides), Wengen or Grindelwald (up one side from the valley, with direct access to the Jungfrau) or Muerren or Gimmelwald (up the other side of the valley, with direct access to the Schilthorn). These towns are all accessed from the Interlaken Ost station.

Your children may also be interested in the Ballenberg Open Air Museum, featuring old buildings and demonstration of old crafts, Swiss traditions, and the like. It's outside Brienz, which in turn is 20 minutes from Interlaken by train.

Harold provides a good summary. Zurich wouldn't be our home base unless it has to be. Many people would split the nights and spend two in Luzern and two in Lauterbrunnen, Muerren, or Gimmelwald. Train from Luzern to Lauterbrunnen takes 2.5 hours, then add more time to ascend the mountains on either side of that valley.

You could take the train from Zurich to Stein Am Rhine. The town looks like a story book scene and sits on the Rhine and is near the Germany border. In the center of the town is an old farmhouse(furnished, and there is a barn behind the house) that is converted to a museum. The train ride is just over an hour and you can see the town in a few hours and get take away picnic supplies and enjoy lunch overlooking the Rhine.

Zürich is a nice city, but not what you want on a first time trip.
Obvious choice is Luzern (aka Lucerne). This is a tourist town first (try not to trip over the japanese and chinese). and you can get spectacular trips on the lake (Vierwaldstättersee, but English-language guide books call it lake Lucerne), which can be combined with trips up mountains.
If you are staying in an hotel, better to stay in Luzern than Zürich.
If you are coming by train, change at Basel for Luzern, if by air thre are direct trains from Zürich airport to Luzern.

Do not dismiss Interlaken. It is also a good base. It is more in the Alps than Luzern (though further from Zürich).
From Interlaken you can do day trips to Grindelwald, Laterbrünnen valley and up the Jungfrau, all by local train.

The railway line from Luzern via Brienz to Interlaken ("Brünigbahn") is very scenic, especially in winter.

The railway line from Luzern via Brienz to Interlaken ("Brünigbahn") is very scenic, especially in winter.

To elaborate on that prior answer, the black and white trains on that line are called the Zentralbahn.