>> Do they have machines at every stop and how do I use the half fare card to get a discount?
Yes, there are machines at tram and bus stops, which have an option for a half-fare ticket or pass (shown as "1/2"). Buy the half-fare option. You won't need to show your Half Fare Card unless you encounter a ticket inspector.
You can try out a simulated Zürich public transport (ZVV) ticket machine here:
>> Also, the trips we make are mostly to and fro (i.e. to the zoo and back), will getting a day pass be useful with a half fare card or should I simply purchase a single trip ticket?
A 24-hour pass costs the same as 2 single tickets, so I regard it as a good idea to get the pass if making more than one trip in a 24-hour period.
You have to take note of which zones you will be travelling in, and buy the ticket or pass to cover the correct number of zones. The zone that covers central Zürich (Zone 110) counts as two zones, and the zoo is in that zone. So to go to the zoo from Zürich Hauptbahnhof, for example, you would need a ticket or pass for 2 zones.
There are zone maps at tram and bus stops. You can also use the online timetable to check which zones your intended trips will cover.
At the top of ZVV timetable page is a map. On the right hand side of the map are zoom symbols (plus and minus signs). Immediately above the plus sign is a symbol like a stack of papers. When you have put your "from" and "to" details in, clicking on this symbol and ticking the box for "Fare zones" will overlay the zone map onto your trip, so you can see what zones your trip will cover. Adjust the zoom to get a suitable view. The attached screenshots show a trip from Zürich Hauptbahnhof (main station) to the zoo (screenshot1) and a trip from Zürich Flughafen (airport) to Zürich Hauptbahnhof (screenshot2).
The trip from Zürich main station to the zoo is all in Zone 110, which counts as two zones, so a ticket or pass covering 2 zones is needed. From the airport to the main station travels through two zones (121 and 110). So for this trip your ticket or pass would need to cover 3 zones - zone 121 being one zone, and zone 110 counting as two zones.
Other ticket options are described here: