<<"Yes, we want to see all 3 places, but at the same time do not want to be
too rush. My thinking of Option 1 is since Rigi is more of sightseeing
mountain and not much of other activities, we can cover it in shorter
time. So we do go Pilatus early morning and spend time until noon. We
should reach back Lucerne by 1pm. From Lucerne, take the boat to Vitznau
and reach by 2.15. That should give us some time to go up and spend
around 1 hr at Rigi Kulm then go down by 4 plus.">>
That will work. And, if the weather does not cooperate, you may do only one of the two.
Keep an eye on webcams. Also, this weather service is my preferred one for details of local weather in Switzerland:
www.meteocentrale.ch/e n/europe/switzerland/w eather-pilatus/details/S06659 0/
If you go early, you may even be back to Luzern before 1pm. Spend some time with the timetable for the Pilatus part. It may be that the return via the cableway to Kriens offers more frequent services to Luzern compared to going down to Alpnachstad and returning by train along the lake shore....just a guess.
On the Rigi, you might consider the option to go down via the cableway from Rigi Kaltbad to Weggis, which requires a 15-20 minute downhill walk from the valley station of the cableway to the ship landing. More than likely, you will then catch the same boat to Luzern that you would have caught if you rode the train all the way to Vitznau.
Here is the boat schedule; watch the dates:
www.lakelucerne.ch/en/ timetable-fares/timetable/ and www.myswissalps.com/bo atlakelucerne
This rather detailed topographic map shows the altitudes. The walk down at Weggis is a 55 meter descent from about 490 meters at the cableway to 435 meters elevation at the ship landing, on good surfaces:
map.geo.admin.ch/?topic=swisstopo&lang=e n&bgLayer=ch.swisstopo .pixelkarte-farbe&catalogNodes=139 2,1538,1396,1430,1436& layers=ch.swisstopo.sw isstlm3d-wanderwege&E=2675874.5 2&N=1209628.02&zoom=10
I turned on hiking trails in that map.
If you wish to see Rigi-Kaltbad, zoom out.
If you look at a map of Lake Lucerne, it might seem that you could go across the lake from Alpnachstad directly to Vitznau or Weggis, rather than going back through Luzern. When I have tried to figure that out, I have found very few connections. With the Lakeboat time table, you can give it a try. But, my bet is that it is faster to return to Luzern then go to Vitznau across the lake, even if it is possible.