Welcome to My Swiss Alps.
By te way, the correct spelling is "Gotthard."
The first three hours from Luzern are by boat.
There are two useful tools to get you started on your question:
This will let you visualize the overall journey:
I have turned on "Traffic" under points of interest. Expand the map and mouse over the icons for transport stops, and you will get a brief list of near term connections.
That will let you see available stops the train route as well as bus and lake boats, and show the timing.
That shows the Lake boat schedules.
From those three you can understand where the stops are , and timing issues. Using the "Aerial View" will give you some idea of the topography.
This site is great for hikes:
map.wanderland.ch/?lang=de&route=all&bgLa yer=pk&resolution=50&X =713045&Y=135730&layer s=Wanderland
Click on the green lines ( trails) and get more information about the trail.
Sometimes it is a little slow, but it is full of information.
Unfortunately, i can't recommend any particular specific hikes or walks, from personal knowledge. I hike a lot around Lake Lucerne, but mostly on the western shores.
The Swiss Path is available along the southern part of lake Lucerne, but much of it is high above the lake.
I think there is one section between Sisikon and Tells Platte along the lake shore.
There are trails in the Gotthard; some high above and parallel to the tracks. I've never gotten off the train....the time passes quickly.
Also, keep an eye on the boat connections at Flüelen on your way north, if your starting point is the Ticino.