Swiss Half Fare Cards: Lucern/Wengen/Zurich

Yes, the Half Fare Card is definitely worthwhile for that trip. If you want to book half-price tickets online, see www.sbb.ch and then choose the type of discount you qualify for from the pull down menu on the second step. These trains don't need reservations and you can also buy the discounted tickets in stations as you go, whether from staff or machines.

As Laura says, you don't usually need to buy rail tickets in Switzerland in advance. If you buy at a ticket counter, show your Half Fare Card when buying. If you buy out of a machine, select the Half Fare option. Remember that Swiss people almost all have half fare cards (they are eligible for a different, cheaper one than tourists), so EVERY ticket machine has this option, including those for local buses and trams. When your ticket is checked, you need to show both the ticket and the Half Fare Card. That's it.

The half price fare is the same as the child fare. Once you select your destination, the ticket machine (touch screen) will have two options "Without Half-fare-card" and "Half-fare-card / Children".
For a demo of how a ticket machine works, see here: https://www.sbb.ch/en/iframes/automatic-ticket-machine-simulation-SBB.html

If you are coming from Austria by train, your half fare card will be valid from the border (Buchs). So you need to buy a split ticket, full fare from your start point to Buchs, and ½ fare from Buchs to Luzern.

You don't "book" rail travel in Switzerland, not in the sense of reserving a seat on a train in advance. You buy a ticket, either at the ticket machine or the manned ticket window. That ticket is then valid on any train that day, you just get on and find any empty seat.