Arno has commented on your plans, and I agree with him. Good thing...he is the expert.;-)
Re: Gotthard Pass
Luzern to Bellinzona takes between two hours 7 minutes and two hours 33 minutes with four trains per hour. The train at xx18 goes to Bellinzona with no changes.
Or, you can change at Arth Goldau. No reason to, but that is a main transfer point for lines for Zürich or Luzern. from the south.
Bellinzona does not have much to see or do at the train station, if I recall correctly. But , it has been many many years since I have stopped there.
If you want to add anice break with a short walk the trip, you might continue to Locarno on the same train, which goes without change of trains to Locarno. 2 hours and 55 minutes. Have lunch at one of the lake shore restaurants near the train station. Ride the funicular not far from the station up to the Madonna del Sasso church, walk up and down a few sets of stairs and admire the view from outside the church, as well the interior of this Romanesque church.
Loarno is more accessible for an easy walk than is Lugano, so I suggest Locarno.
That is probably an 8 hour to 9 hour round trip, but you get to see the Gotthard Pass twice. Well worth it in my opinion. Watch for the church at Wassen.....the train passes it 4 times on each trip. ;-)
Re: Hikes near Luzern good for a 3 year old. That is a tough one - I don't really know how much stamina a 3 year old has. Remember that the Swiss don't notice hills when they walk. ;-) A topo map can give more information:
map.geo.admin.ch/?topic=swisstopo&lang=e n&bgLayer=ch.swisstopo .pixelkarte-farbe&X=183840.00&Y=67 1580.00&zoom=5&layers= KML||https:%2F%2Fpublic.geo.admin.ch%2FuMAIJrFBQ-2EBw2HqNSsNQ
In Luzern, you can walk along the lake shore path to the Transport Museum ( Verkehrshaus.) Great views across the lake to mountains. About 2 1/2 km. walk from the main station. Served by bus and train and lake boat stops if someone gets tired:
There is a famous trail around the southern part of Lake Lucerne. The "Swiss Path" (der Weg der Schweiz) follows the shore line of the lake from Rütli Meadow ( boat stop) up quite a hill to Seelisberg, then on around the lake to the end at Brunnen. I don't think any stretches are suitable for a 3 year old and also interesting for scenery.
However, this one near Seelisberg might work:
Take the lake boat to Treib, then the funicular up the hill to Seelisberg. Then walk up hill for about 600 meters or take the bus to the grand old Hotel Sonnenberg (now run by an Asian guru). There is a promenade at the hotel and trail partway back to Seelisberg near the cliff edge ( safe, good railings) with spectacular views across the lake and mountains.
You might do it as part of a lake boat trip, or, take the train to Brunnen and then the boat across the lake to Treib. The harbor in Brunnen is about a 10 minute walk from the train station...there is a shuttle bus at least some of the time.
In Luzern, take the kids to see the Lion. Maybe a km from the station. I'll be surprised if it does not impress them and puzzle them. My 8 year old granddaughter spent a lot of time just looking before asking questions. I have attached an image of a page from a photobook of our trip to Luzern in 2009.
They'll quite possibly enjoy the nearby Bourbaki Panorama..the 8 year old did and the adults always do.
If you go up on Pilatus, consider taking the lake boat to Alpnachstad, then the cograil train up the mountain. (The train can get you to Alpnachstad faster than the boat, but the boat is more fun.)
Come back that way or by the cable car to Kriens, thence to Luzern...bus, I recall... The cograil is one of a kind; I prefer it. I have attached some really old photos; sorry for the technical quality.
When you leave Luzern, you can go to Zurich directly ( via Zug...a 45 minute train ride). or, you can take time to go through Bern. The walk through the old town down to the bear pits is interesting because of the fanciful fountains in the streets....including one of an ogre ( der Kinderfresser) , the old clock tower, and the bear pits by the Nydegg Brücke.
That might interest the kids more than Zürich.