Train from Lucerne to Milan

Define "best". Rail websites will always choose the fastest, which from Luzern to Milan will be through the new Gotthard tunnel under the Alps, the longest in the world. But you will still see mountains, as you go along the valleys either side of the Alps.
Other routes are possible, but will take hours longer.
Option 1: via the older, shorter Gotthard tunnel.
Option 2: "over the top", via Andermatt and a bus over the Gotthard pass.
Option 3: Longest and most scenic, via Chur, the Bernina Railway to Tirano, then onward to Milan.

To look up trains, go to the Swiss Railways website: https://www.sbb.ch/en and enter Luzern to Milano Centrale, with date and time, then click on "Search". You can buy tickets on the same website.
Hint: Click "Reserve seats only if required", most Swiss trains you cannot reserve seats.

Thanks so much. Ideally we would want a trip which includes scenery. I've also been considering staying at Lake Como instead of Milan - as I believe that Milan might just be a little too big (and some say dangerous, pick pocketers etc) for us.

This is probably all in the same kind of direction though as the train that takes you under the alps right? Chances are that we would like to go on top considering the scenery we'd experience from the train that way.

Do you possibly have any other suggestions of where to stay around the south border of Switzerland to break the journey between Lucerne and Riomaggiore?

I took the train from Luzern to Como on two different occasions. I don’t know if there are alternate routes to take but I remember the scenery was breathtaking. If you don’t like big cities, stay in Como instead of Milan. The city is compact, easy to walk to all the main sights, loaded with terrific restaurants (Il Pomodorino on Via Cinque Girante or I’ll Pinzomonio on Via Bonanomi are both very nice), beautiful waterfront.

For a spectacular view of the Alps and Lake Como, take the funicular up to the village of Brunate.

Como is a big lake, with two forks (shaped like am upside-down Y), so you need to specify where on the lake before deciding a route.
Como (the town) is on the main direct route from Luzern to Milan. It is the first town in Italy, just south of Chiasso last Swiss town). From there you can get boats to anywhere on the lake.

Another popular place is Varenna which is halfway along the right (east) side of the lake. This has the advantage thta it is in the railway line that comes over the Bernina Pass. If you haven't heard of the Bernina Line, look it up, the railway line is a World Heritage site.
You then take the long rail route from Luzern: Luzern - Chur - (Albula Pass) - (Bernina Pass) - Tirano - Varenna.
If you are a map person, this rail map of Switzerland might make this explanation easier: https://www.sbb.ch/content/dam/infrastruktur/trafimage/karten/karte-sts-pass-gueltigkeit.pdf
After your stay in Varenna, continue by train via Milan to your final destination.

Should not listen to "Some says ......" because generally the "Some says" know very little. If you are avoiding Milan because it is big city, that is OK. But it is silly to avoid it because it is "dangerous." Because -- it is not dangerous at all. Milan doesn't get a fair rap as a tourist site but it has plenty to offer from the opera house to the Last Supper.

To maximize scenery, you can take the Gotthard-Panorama Express that starts with a boat trip from Luzern to the far end of the Vierwaldstaettersee at Fluelen, then transfers to a train over the Alps via the old Gotthard Tunnel and ends in Lugano where you can get trains continuing on to Como and Milan. You need 1st class to ride in the panorama cars.
https://www.sbb.ch/en/leisure-holidays/ideas/offer.html/zuege/gotthard-panorama-express

Choose your itinerary for the all the reasons above but please do not call Milan dangerous as it does not deserve that label at all.

If you take the Bernina route as outlined by Chris you will go over the alps, not under them in a tunnel. It is the best for scenery. And you can overnight in Varenna, which is less expensive than Milan (in our experience) and still not a bad journey to Cinque Terre. We did that journey a few years back: it is one hour from Varenna to Milan, then catch the IC train that goes directly to Monterosso with no changes. I believe it departs Milano Centrale just after noon but you chould check.

We used the Bernina route to reach Venice from Zurich two years ago and loved it. We added an extra overnight in Switzerland because it was after our flight from the US, departing Zurich in the afternoon. For you, departing from Luzern in the morning, you can easily reach Varenna. We had a Lakeview room with balcony at Alberto Milano for considerably less than anything I could find in Milan. (We do like Milan, and had a nice stay there when going to La Scala for balllet, but preferred to stay in Varenna this trip).

Thank you so very much for the very detailed replies. I really do appreciate it and yes, I apologise for being too sensitive to what others have said about Milan and should rather formulate my own opinions on it :) So sorry about that!

I like the idea of the Gotthard-Panorama Express that ends up in Lugano where we could then continue onto Milan or Como. I can't seem to see where to book the Gotthard-Panorama Express trip online though. Does one need to book? We'll be in Luzern for 2 days before departure. Would I be able to book then and is it advisable to book beforehand or would I be okay booking a day or two before? It's only my wife and I travelling.

Thanks for all the great advise.
A group of four of us are planning a very similar trip at the end of September.
Would it be advisable to purchase our tickets upon arrival, our should we pre-purchase on line?
Do tickets cost more closer to the travel date?
It is very important to my friend who is 81 we take the train with the glass roof.