You can plan your trips using the timetable:
This map is useful, too.
By using the menus on the map you can make public transport stations appear or disappear, see the locations of accommodations, and search for hotels by name.
For your departure, my choice would be to stay in Luzern and not transfer for one night in Zürich. If you wish to see the Rhine Falls, you can do it from Luzern with only modest extra inconvenience, plus train times. And, since you will be spending a lot of your time on other days in Luzern on travel out of the city, you might decide that sightseeing around the city of Luzern or a boat ride to the southern end of the lake or a hike nearby would be more enjoyable than seeing the Rhine Falls.
The benefit is that you do not use your time to move to another hotel and do not need move your luggage in and out. Personally, I find moving from hotel to hotel wastes time that I would rather use on sightseeing.
Trains from Luzern to Zürich airport run twice per hour.
The ones at xx35 take 1 hour and 7 minutes with change of train at Zürich main station. If you miss the connection in Zürich for any reason there are about 5 or 6 trains per hour from Zürich main station to Zürich airport.
The trains at xx07 go direct from Luzern to the airport with a stop in Zürich main station and take one hour and 3 minutes.
See attached screen grab
Since you would be traveling during the morning rush hours, you can expect that the trains will be rather full, so I'd try to get on as soon as one pulls into the station in Luzern. And, you might expect some very slight delays on the busy stretch between Zug and Zürich main station.
Not sure what time you want to be at the airport, but, for peace of mind about a transfer at Zürich main station during rush hour, I would try to take the direct train (xx07) even if it meant that I'd have to leave a bit early. If you need to be at the airport two hours a head of your departure time, the train at 0810 could do the job.
Your airline can guide you about timing at the airport.
If you miss the connection in Zürich for any reason there are about 5
or 6 trains per hour from Zürich main station to Zürich airport. However, there are several levels of tracks in Zürich main station, and getting from one track to another might take anywhere from 1 minute to 8 or 10 minutes, depending on your familiarity with the airport. There are good signs.
In the timetable, if you click on the "+" sign to the left of the individual journey, you will more details including see track numbers for connections and time allowed. In Zürich main station, tracks #3-18 are on the "main" station level and easy to transfer between. You can move from platform to platform at the railhead, or at an underground passage about halfway out under those platforms.
Tracks other than 3 to 18 take a few more minutes to get to, and can be on different levels. The luggage carts can go up and down escalators. Although they are free at the airport, at the main station you need a 2 CHF coin to free the cart from a storage queue. You get it back when you plug the cart back into a different storage queue near where you are finished with iit.