Lucerne to Mt. Titlis -need help planning transportation options

I think it will be pretty easy on your own. The Eurail pass covers direct trains to/from Engelberg and gives you a 25% discount on the Engelberg-Titlis lift. Show your pass and buy that ticket on the spot.

[Edited] Mt. Pilatus does sometimes include more pieces and is often done in a circular route, such as the Golden Roundtrip that is partly covered and partly discounted with a pass. However, the steep inclined train to/from Alpnachstad does not run in winter. Whatever you choose to do around here, a Swiss ticket agent can sell you just the pieces you need to supplement the pass.

An advantage to doing this yourself is that you won't have money already spent if the weather turns out to be really unsuitable for the trip.

as it looks like there are a lot of ways to get there

Lucy, not really, there is really only one route to Titlis, train from Luzern to Engelberg, about 10 minute walk to the foot of the cable car, then up the cable car (actually a series of 3 connecting cable cars) to the top. No tour needed.
If you already have a pass that covers you to Engelberg, just get on one of the hourly trains from Luzern to Engelberg, and buy your cable car ticket at the cable car station, showing your pass for a discount. Otherwise buy a ticket at Luzern station all the way from there to Titlis summit.

but I am not sure what other passes to buy

Only ever buy a pass after you have costed normal tickets and decided a pass is cheaper.

Note: 21st December is the shortest day of the year. Start out early or you will run out of daylight.

I was there this summer from Lucerne and used the train, so simple.
There is a free shuttle to mt titlis entrance And circle around tiny Engelberg.
Watch weather carefully bef u leave lucerne. Mt Pilatus and Mt Rigi are close by as an option if weather is iffy at Titlis.