In addition to what Arno wrote, there is a bus that runs along the lake shore from Brunnen to Küssnacht am Rigi. It connects with the train station at each end. it is likely faster than the boat, but less scenic.
This map will help you:
I have used the menu under Points of Interest to turn on Traffic and checked train and bus stops, so you can see them on the map. Zoom in and out.
This link to the map shows that your hotel is halfway between two bus stops (Weggis Dörfli and Weggis Parkseehaus). It is about 450 meters from the Schiffstation, uphill:
map.search.ch/Alpenbli ck-Hotel-Restaurant,Weggis,Luze rnerstr.31?poi=haltestelle,zug&po s=675494,209486&z=256
Lake Boats -
The boats to Weggis indeed can be reached from the train station at Flüelen (close to boat dock) or at Brunnen ( 10-15 minute easy walk ors huttle bus from train station.
Coming the other direction ( southbound) on the boats, certain boats from Luzern stop at Weggis. The ship landing at Luzern is a 2-3 minute walk from the railhead, out the main doors and across the plaza.
This link gives all the boats on Lake Lucerne ( der Vierwaldstättersee):
Note that there are several routes on the lake.
There is a map somewhere on that site.
To understand the timing and station stops (and even route maps, down in the menus after a journey is selected) use the timetable. It will get you from Germany ( e.g., München) to any public transport stop in Switzerland, including bus stops. It includes all public transport in Switzerland, including boats, cableways, etc.
Unless forced by selection of a "Via" or "Connection" it defaults to the fastest route.
It has a LOT of information and can take a while to understand the details buried in the menus.
Please read the instruction at this link. They are very useful:
I have attached a screen grab for a trip from München Hbf to Weggis Dòrfli bus stop near your hotel. ( I misspelled Dörfli in the title of the file....ooops.)
<<"Also can anyone recommend anything that would be great to do for a day? Thank you!">>
What are your interests? When are you traveling?
Luzern has a lot:
Apart from the obvious and famous, one of the best museums in the world for Picassos and Klees is the Rosengart collection, 3 minutes from the main station.
The tourist information office by track 1 at Luzern main station is very helpful and the staff really know their stuff.
The Rigi -
The Rigi is easily accessible, although it is a 10-15 minute uphill walk from your hotel to the lower station ( "Talstation" (valley station) ) of the cableway ( Luftseilbahn, LSB) The LSB goes up to Rigi Kaltbad-First. It connects with the cog rail train from Vitznau to/from Rigi Kulm. (Kulm is related to the English word "culmination" and means the top or the finish or the end. Usually, the top of a mountain .)
Are you hikers? Ages of your party?
Some nice pictures toward the end in this link:www.myswissalps.com/fo rum/topic/the-swiss-path-and-southern-lake-lucerne
I've attached a couple of pictures of the lake shore one shows Weggis at the foot of the Rigi) taken from Hammetschwand, on Bürgenstock, as well as one one of Bürgenstock from Luzern.