Lucerne on one's own

The train from Zurich to Luzern takes a little more than an hour, so you could easily do it as a day trip from Zurich. Engelburg is also an easy day-trip from Zurich, about 1.5 hours by rail.

If this is a leisure trip, not business, then you would be closer to the action by sleeping in Lucerne, instead of Zurich.

I also recommend staying in Lucern instead of Zurich if you can. It is a beautiful city on the lake where you can take a leisurely cruise, visit the Lion Monument, take a walk over the Chapel Bridge to the Jesuit Church and just take in the city. Mt. Pilatus is a great side trip (cog railway up the mountain, cable cars down the other side) as is Mt. Rigi and if you are interested in trains, visit the Swiss Museum of Transport. Great sights, great food, relaxed atmosphere.

I would also suggest spending at least one (and preferably more) nights in Lucerne. It's a beautiful city and there are a lot of possibilities to choose from. It's difficult to provide any information on how to "do it on my own" without having some idea of your interests or the type of sights you want to see.

Thanks for all your replies.
Where n how can one buy the Mt Pilatus tickets? Is it a part of Lucerne?
Thanks

Pilatus is just outside Luzern, Luzern's suburbs extend to the foot of the mountain.
You can buy tickets to the summit of Pilatus from whichever station you are starting from (Zürich, Luzern, ...). You can do a round trip, take a boat or train from Luzern station round to the "back side" of Pilatus, go up by funicular, then down the other side by cable car to Luzern, and a city bus back to the station.
Details here: http://www.pilatus.ch/en/railway-cableways/golden-round-trip/
"Silver" or "Gold" options are train or boat.

Note that the month of the year, and daily weather is important in Switzerland. In particular, you might not want to pay for a mountain ascent until you have either read the visibility forecast or looked at their summit webcam display online. OTOH, it sounds like you have only one day, chosen in advance, to do this. That is not uncommon, I know, but it's not ideal for a mountain visit.

1) The train from Zurich to Lucerne does NOT take a little more than an hour. It's actually between 45 minutes and 58 minutes.
2) Do stay in Lucerne if possible. It's much prettier and interesting than Zurich
3) For any trip you want to take (Pilatus, Rigi, Titlis) you just walk up to any train ticket window and tell them where you want to go.