The Swiss Half Fare Card and the STS Family Card are not valid for the Italian leg of the trip, but only for the Swiss leg (that is, Lugano-Chiasso). Please review the map we link to at the Swiss Half Fare Card page (under the header 'Features') for an exact overview of the validity network. I found somewhat different costs, but these may depend on the exact connection you choose. The quickest and easiest connection I found (a direct connection without changes by EuroCity, taking about 1h00) costs CHF 21.00 per person for 2nd class, including the 50% discount for the Swiss leg. If you want to take the 50% discount for Switzerland into account when checking prices, you should select the option 'Half-fare travelcard' under the header 'Reduction for' during step 2 of the process.
All you need to know about traveling by train, bus or car in Switzerland.
Hand picked Swiss hotels and other accommodations are right here.