Lucas gave you excellent answers, but I would like to add a few more comments.
<<"what is the airport called (Kloten?) ">>
It used to be called Kloten, but I am told that is a vulgar word in some European language...Dutch? Swedish? Don't remebr which. Kloten is the name of the village nearest the airport. But, a few years ago they tried to rename it "Unique." I don't think that stuck - at least I don't see it often - so I just call it Zurich Airport in English, and Zürich Flughafen in German.
<<" I understand that there are trains from the airport direct to luzern leaving at 47 mins past the hour.".
True. The xx47 does stop in Zürich Hauptbahnhof ( main station) but goes on to Luzern with no change of trains. The train from the airport at XX18 also is good for Luzern. It does requires change at Zürich main station, but you have 8 minutes to make that...quite possible...standard minimum allowance is 4 minutes.
<<"Is there only one train station in luzern?">>
Lucas answered correctly. There is only one station named Luzern. Station naming is unambiguous. Maps and timetables require exact names, and al must be different. Even in in Zürich, where there a few train stations here and there as remnants of the old lines coming into the city to a terminal, those stations have names such as Zürich Teifenbrunnen or Zürich Stadelhofen.
<<" ..Is the station near the CHapel Bridge? I will be staying in that area and I dont want to walk a long way to Old town area with my luggage">>
Lucas answered that one as well. What he did not mention is that the Chapel Bridge is in the Old Town, connecting parts of the Old Twon (Altstadt) on each side of the river.
Take a look at a map...it will help you understand the locations. This one is useful:
Zooming in on Luzern:
at zoom level 32, the rail lines become very clear:
and zooming in further, especially if you use the aerial imagery, you can see the main station, the Chapel Bridge, the Old Town, etc.
<<" Train tickets I assume we can buy at the airport? ">> Yes.
<<"the rest will be sightseeing the beautiful city of luzern and zurich of course.">>
Zürich is not an "of course." Luzern is. Zürich is only an "of course" if you have specific known interests there. Shopping? Specific museums?
Luzern is much more scenic, and has some superb museums.
It is not as scenic as other possible destinations.
We don't know your arrival and departure times, but if they permit, I'd consider spending time in the Jungfrau region instead of Zürich. seeing the region is possible as a day trip from Luzern, but if you go all out and spend the money to go up to Jungfraujoch, it become a very long day as a day trip. . Check the timetable that Lucas linked for you.
I'd consider staying in Interlaken ( good train connections but not so scenic) or, much better, one of the Alpine villages south of there -Wengen, Mürren, Lauterbrunnen, Grindelwald. Search all those names in this website for more details. It is ski season, so they may be full, but, I'd try. The atmosphere is very Swiss, and the mountains covered with snow are magnificent.
Interlaken Ost itself is 2 hours 12 minutes to 2 hrs 16 minutes from Zürich airport, twice per hour.. But, the villages are accessed by slow mountain trains into the valley from Interlaken Ost, and add 25 minutes to almost an hour to go into or come out of the valley, depending on which one you choose. So, you might want to stay in Interlaken, and take a day trip or two into the Lauterbrunnen Valley and up on the mountains - many options. Maybe go up to Schilthorn, for example:
Just some ideas for more snow. ;-)