Zurich to Lauterbrunnen trains

By train service, do you mean route? I have heard that you can take a route that goes through Bern or a route that goes through Lucerne, and although the first is a little faster, the second is more scenic.

Use the Swiss rail site sbb.ch/en and it will show you the best routing options, if best means most efficient. I think you have to change trains in Interlaken Ost no matter what route you take.

The Swiss rail site (sbb) will always show you the fastest way, which is through Bern, but a slower and more scenic route is via Luzern with the beautiful Luzern-Interlaken Express over the Bruenig Pass. To find it, just enter Luzern in the "Via" box.
https://www.sbb.ch/en

The Swiss network of trains, postbuses, gondolas, boats, funiculars and other modes of transport is highly organised and fully integrated, and really well timed. That's regardless of if it is the Swiss Federal Railway (abbreviated as SBB/CFF/FFS because of the three main languages) or one of the private or semi-private companies running various transport. Things are timed to fit together like cogs in one of the famous Swiss watches.

There are basically two routes from Zurich airport to Lauterbrunnen. I'm pretty sure that Rick outlines them in the Switzerland guidebook, at least he used to.

What is best for you, only you can say. I will lay out the two routes and you can choose.

One is Zurich Airport to Zurich main station known as Zürich HB (HB means Hauptbahnhof, literally main train station), to Bern, to Interlaken Ost where the train ends and you need to walk over to the narrow gauge train up to Lauterbrunnen. You may need to change at Zürich HB, or at Bern, or at Spiez which is where the line to Interlaken branches off the main line. This will depend on what time of day and month you travel, and what is available at the time. You definitely have to change onto the narrow gauge train up the hill. Be careful because it will split part way and one half will turn left towards Grindelwald and the other half will continue up to Lauterbrunnen where it will conclude its journey. It will be well marked, and there are illuminated signs on the train carriage doors saying which is which. There are two stations in Interlaken. The first stop will be Interlaken West, you need to go 3 minutes further to Interlaken Ost. Sounds complicated but it isn't. Lots of people will be doing the same thing, and the announcements on the trains are in several international languages including English.
That will take between 2:08 and 2:18 as far as Interlaken Ost depending on where (Bern, Zürich HB, Spiez or a combination) you need to change.

The other option is to go via Luzern (Lucerne in English) and a narrow gauge cogwheel train over the beautiful Brünig Pass from Luzern to Interlaken Ost. The train turns around in Meiringen where Sherlock Holmes had his fight to the death with Moriarity at the beautiful Reichenbach Falls. The train turns but you don't get off. At Interlaken Ost you still have to change the train described above. The direct train from the airport alternates every half hour with the train requiring a change at Zürich HB on the way to Luzern. It takes 3:07 or 3:17 to Interlaken Ost depending on the change or not at Zürich HB. So all in all it is about an hour longer this way which is made up by fabulous scenery of waterfalls, lakes and cogwheel.

Whichever way you get to Interlaken Ost the train up the hill to Lauterbrunnen takes just over 20 minutes.

So which wins for you? Scenery, mountain pass and slightly slower or less scenery, flatter and faster? Your choice.

Sam and I were writing at the same time, without seeing each other's answers...

I might add that if you are traveling on point-to-point tickets, the route via Luzern is cheaper than via Bern. Via Luzern in 63.60 CHF full fare, while Via Bern is 81.60 CHF. That is because Swiss rail prices per km traveled, and it is shorter through Luzern than Bern. Through Bern is faster because the tracks run through valleys so are straighter and the trains run faster. The route from Luzern to Interlaken is over mountains so it has lots of curves and stretches where the cogs have to engage the rack for steep climbs and descents.

Definitely take the scenic route—it is breathtaking and well worth the extra hour.