How to get from Lucerne to Varenna?

Several ways,
1. Train to Lugano, then a bus to Menaggio, then ferry to Varenna.
2. Train to Como S. Giovanni station, bus or taxi to the water front, ferry to Varenna.
3. Train to Milano Centrale station, change to train to Varenna.

Take your pick. No 3 is the easiest, not necessarily the cheapest or fastest.

or

modify "3. Train to Milano Centrale station, change to train to Varenna." to read

3a. Train to Milano Centrale station, but get off at Monza; change at Monza to train to Varenna.

Thanks for the help!

One last question on this topic. Is it recommended to purchase train tickets in advance or can this be done the day of? Would the ferry be the same? Thank you!

Depends on which of the 3 ways you are going.
Looking at June 9 date for example.
1. Not too much savings with advance purchase. There is a direct train from Luzern to Lugano about 8 am. 61 CHF reg price 2nd cl, vs 52 CHF Super Saver (nonrefundable).
2. Some savings with advance purchase. Train about 8:30 with a change at Arth-Goldau. 76 CHF reg price 2nd cl. vs 45 CHF Super Saver Smart (nonrefundable).
3. To Monza per Nigel. Some savings with advance purchase. train about 8:30 with a change at Arth-Goldau. 80 CHF reg price 2nd cl. vs 47 CHF Super Saver Smart (nonrefundable). Trenord fare from Monza to Varenna is 5.60 EUR.

Learn about Rome2Rio - a website which can help in route planning for many types of transport.