<<"Would it be advantageous to get the Swiss Half Fare Card since we will be traveling by boat to and from Luzern from Vitnau and possibly by rail to Interlaken or Bern on one of our days?">>
Rockoyster has given you a very complete answer, including the correct way to cost out rail travel options.
I'll add a guesstimate.
Based on what you describe as your plans, especially if you go up to the Rigi and/or Pilatus, or add the trip to/from Interlaken, my guess is that the Half-fare card at 120 CHF will probably pay for itself. However, it will probably be close, that is, not a whole lot better. If you were to include a bit more long range travel for a few more days, I'm sure it would save you money. It also will give you a roughly 50% discount on local trams and buses, and on a day card for Luzern, if you wish to use the city's buses.
The real treat at Interlaken is to go further in the valley to Lauterbrunnen take the cograil train ( which will be running) at least as far as Kleine Scheidegg, returning via either Wengen/Lauterbrunnen or Grindelwald.
With your case in Vitznau, I might tend to drive to Lauterbrunnen to save some time, then take the cograil train. One hour 15 mintes from Luzern, but more like one hour and 35-45 minutes from Vitznau (Check the timetable after October 18th, when the new edition comes out). Adding the trip on slow mountain railways makes a very long day. Public transport from Vitznau takes about 2 hours 15 minutes to Lauterbrunnen, and the journey over the Brunig Pass is reasonably scenic.
After October 18th, when the new timetable is available , check the timetable for November 4th, and the transport to Schilthorn up from Lauterbrunnen. They shut down for maintenance in November, but I thought it was later in the month. If it is running, that would be quite adventure. You could even drive as far as Stechelberg and park near the base of the cableways.
Bern is closer and nice for a short day trip. Once per hour, one hour trip, on the hour, from Luzern..
Trains at intermediate times taking a bit more time run several times per hour.