Train Basel to Lucerne

Go to http://www.sbb.ch/en/home.html for the detail. That's the Swiss Railways (SBB) official website, in English.

There is a frequent service to Luzern (the German spelling and what is used locally) from Basel. You will need the Basel SBB station, not the Basel Bad station. You will tote your own luggage.

Both Basel SBB and Luzern have street level platforms, but at Basel SBB you will need to use the overhead bridge to reach the platforms as it is a through station. There are large wide escalators and an elevator to each platform. At Luzern you just walk off the front of the platform right into the station. If you go down to the underground passage you can avoid traffic as you walk to your destination. Again there are escalators and elevators.

If you are going to a ship in Luzern or want a taxi you don't go downstairs, just outside.

The fare doesn't vary, you can get tickets from machines or humans, they both speak English. The fare for adults is CHF32 pp one way. You can take just over an hour no change, or with one change a few minutes more.

You cannot pre-book trains in Switzerland. They all operate on a turn-up-and-go system.
The price is the same whether you buy the ticket in advance, or from the ticket machine or the ticket sales window on the day, CHF 32 full fare second class single, CHF 16 if you have a half-fare railcard.
Basel SBB to Luzern there are 3 trains per hour, and it takes just over 1 hour.
Just turn up at the station whenever you like, buy 2 tickets from the ticket machine and look up at the departure board for the next train.
If you look at the photo here you can see the departure board. About 2/3 of the way down you can see the 15:03 departure to Oltern, Luzern, Belinzona and Lugano from platform 7. That would be your train.