train or rental car

You can't drive to Mürren, so it's an easy answer in my opinion.

If you drive , you will have to ditch the car at the carpark in Lauterbrunnen . Getting to Lauterbrunnen from Zurich by rail is far more comfortable than driving, so do as Badger says .

Definitely take the train. A car will be of no use to you once you arrive in Lauterbrunnen. You can catch the train right at Zurich’s airport ( “Zurich Flughafen”) and be in Lauterbrunnen 2:40 later. There may be a change of trains in Bern and there will be a train change at Interlaaken Ost (East) station. You’ll probably want to get a train pass for the Berner Oberland because you’ll. be using the gondolas, cable cars, cogwheel trains and local trains to get around and the costs add up quickly.

The Berner Oberland Pass covers just about all transportation for 240 CHF for 3 days and 280 CHF for 4 days. ( Other timeframes are also available). It covers the train fare from Bern or Lucerne to about everywhere in the Berner Oberland. It covers the cable cars from Stechelberg to Murren and the cable car from Lauterbrunnen to Grutschalp along with the train from Grutschalp to Murren. A great hike is going up to First Glacier and the cable car from Grindelwald to First Glacier is included ( Just this ticket alone can cost 72CHF roundtrip). The gondola to the Schilthorn is 1/2 price and priced at 42 CHF Roundtrip.
The Jungfraujoch train is discounted to 99 CHF. The Berner Oberland Pass covers trains running from Bern and Lucerne to the Berner Oberland. So, the only train ticket you would need to add is for the train from Zurich Flughafen to Bern which runs about $35 one way. You can buy the Berner Oberland Pass online in advance of your trip, or at the airport in Zurich when you arrive. Check www.BerneseOberlandPass.ch for more details.

Have fun in Switzerland!