Most of the regular passenger boats (not the special dinner boats, etc.) operate from the main dock directly in front of the Luzern train station, and follow standard routes. You can buy tickets, get timetables, check routes at the main dock ticket windows facing the train station. You can check schedules, and I presume buy tickets, via the SBB website. You can buy tickets at the station or dock prior to departure, or in advance. We were in Switzerland for a longer period of time and had a Swiss Travel Pass so we never bought individual train, boat or bus tickets. We did have to buy discounted tickets for Pilatus & Titlis.
Before going on any mountain excursion, check the weather cam for that mountain. All the popular mountains have weather cams, and the weather at the top can be different than what you encounter in town. Two years ago we had an overcast drizzly morning in Luzern but the top of Pilatus was in brilliant sunshine, so up we went and it was gorgeous. We've also ridden through extensive clouds on our way up to Pilatus and Rigi only to emerge into brilliant sunshine. I prefer to go in the morning but that's just me, it really depends on the weather at top and the forecast.
Rigi has very little infrastructure at the top, is more of a rounded hilltop with trails, although you do go through several small station stops along the way. The lakeside ("red train") is more scenic than the other "blue train" side. Pilatus is more craggy, has a diverse infrastructure - two hotels, large outdoor plaza with restaurant service from one of the hotels, indoor hall with snack bar, chairs, gift shop, etc, - very little hiking or trails, and nice views of Luzern on one side and the mountains & lake the other way. Titlis is typical high Alp scenery, snow, snow sports, and smaller but decent infrastructure. It is most like what you'll encounter in the Jungfrau region. I wouldn't pass it up just because you'll be going to the Jungfrau region later, but if you go to Rigi & Pilatus you could skip Titlis. Your mom may not like the cold, snow, or altitude.
We spent almost all our time based in Luzern on our two previous trips, so I don't think three days is too long. We spent some days in Luzern itself, but also took several partial or full day trips on Lake Luzern, small towns on or around Lake Luzern, and some to the Jungfrau region. Look at the links on this website for things to do in Luzern.
I'm not familiar with that hotel, so offer no opinion of it. We stayed close to the train station and boat docks, which was certainly convenient. However, there are an abundance of buses & bus routes in Luzern, and they all seem to go to or depart from the train station. Our hotel provided us a city bus pass during our stay, although we didn't need it since we had the STP.