One day from Zurich to Interlaken

"Best way"? Fastest or most scenic? Fastest way is thru Bern, most scenic thru Luzern.
What should we see? Well, there is the Eiger, the Moench, and the Jungfrau. A day trip will not even scratch the surface of the Berner Oberland.

You say your "ancestors came from the Interlaken area", do you know precisely where? If so you could go there. Interlaken itself is often denigrated, but merely because it has a lot better nearby, the Jungfrau region with world class mountains.

"What is the best way to get there and what should we see?" - The best method of transport is by train, as is the case for getting round much of Switzerland. The quickest route is via Bern. The most scenic route is via Meiringen and Luzern, over the Brünig pass.
You can look up train times on the SBB (Swiss Federal Railways) website: https://www.sbb.ch/en
This will by default find the quickest route, i.e. via Bern. If you want the Luzern route, you need to add "via Luzern" to your search.
But also note the Luzern - Interlaken route is closed due to maintenance until 30/4/2018 and there is a bus instead.

The construction work on the Luzern-Interlaken rail line is scheduled to be complete by April 30, so you should be fine in May.

It's best if you have time to see the mountains close-up by getting up to the mountains (via Interlaken Ost and then via Lauterbrunnen) that Sam mentioned. But if you don't have enough time for that, you could visit Harder Kulm, which is up above the city of Interlaken. A funicular takes you up to the outdoor cafe. It has a cantilevered overlook platform that has a clear floor, and the view is of the city and lakes below, with the snow-capped mountains, beyond, in the distance. It's impressive, although not so awesome as getting closer to the mountains through the Lauterbrunnen Valley and near Kleine Scheidegg and Mürren and Allmenhubel.

Definitely take the scenic train route from Zürich, through Luzern, the Brünig Pass, and Meiringen, called the Luzern-Interlaken Express, on your way, as Chris F mentioned. It takes you past beautiful mountain lakes surrounded by chalets, and waterfalls off cliffs.