Train from Lucerne to Paris

The train that goes to Paris.

Look up connections and other information at www.sbb.ch

make1485, all of the websites that provide train schedules will show you the most efficient routes if you just put in the departing station and the destination. You dont have to figure out a routing or connections yourself. Unless you mean something different by "best".

Looks like it's about a one hour train ride from Lucerne to Basel, using the Swiss rail website that Badger provided.
Once in Basel there are several trains per day to Paris - taking about 3 hours. Research that one on the French railways SNCF website: https://www.sncf.com/en
There will be a change of trains (and train lines) in Basel so you'll want to build in enough of a delay to give yourselves plenty of time to make the changeover.

There will be a change of trains (and train lines) in Basel so you'll
want to build in enough of a delay to give yourselves plenty of time
to make the changeover.

If you're buying two separate tickets you need to do that. But why would you buy separate tickets?

You do not need to build in a buffer, not even if you travel on separate tickets. Connections between SNCF and SBB and vv. are always protected.
If you book this on www.sbb.ch you will get two tickets anyway, one for the Luzern - Basel part, and one for the Basel - Paris part. The Luzern - Basel ticket will however be a special "connection" ticket that is cheaper, as it is part of an international journey and thus not subject to VAT.

SNCF cannot book the whole trip as they cannot book Swiss domestic tickets.