Need help on trip ton Switzerland, Paris and London

It's a bit easier to understand if you talk in "nights"

You have 15 days means you have 15 nights? Or you do only have 14 nights?

I can answer some of the questions, and give my opinion on others.

Day 1: Arrive Zurich around 9am and take train from Zurich to Lucerne (any suggestion of place to visit after check in hotel?)

Just walk around and see the covered bridges, the painted buildings, the Lion Monument, etc. Luzern is very pretty, and you'll probably be too tired to do anything but walk around, which is the best way to fight jet lag.

Day 3: Lucerne to interlaken (will it be better if i stay in interlaken or other place like grindelwald)

You want to stay IN the mountains, not near them. Stay in Lauterbrunnen (in the valley), in Wengen (up one side of the mountains) or Gimmelwald or Mürren (up the other side of the mountains). Only stay in Interlaken if you want a casino, a sex shop, lots of jewelry shopping, large hotels, and lots of tour groups.

Note that your plan to have 2 days in the Berner Oberland (the mountain area near Interlaken) and one day in Zermatt is very risky. Mountain weather is very unpredictable and very changeable. If you get to Zermatt for your one day and it's cloudy, you've come a long way and spent a lot of money for nothing. I'd pick one of these two areas and spend all three days there. I haven't been to Zermatt, but I spent four nights (three full days) in Mürren and had a great time; there's lots of walks, hikes, and lifts there, so you won't run out of things to do.

I'm not a shopper, so I probably shouldn't comment on this, but I see you have allotted TWO WHOLE DAYS to shopping centers. Even if you do want to shop, I'd pick only one shopping center. It makes sense to do the one near London, as that's at the end of your trip, so you don't have to carry your purchases all through the trip.

Any recommendation of the hotels for the the places I stopover? I am actually have Budget on my trip.

We need to know your budget range in order to give recommendations.

Harold has given you excellent advice about Interlaken and Zermatt. It takes a long time to get to Zermatt, and if it rains...all in vain. There's a week's worth of stuff to do in the Lauterbrunnen, Murren, Grindelwald area.
Remember that you'll be in the mountains in October, so dress warmly.
By the way, in Lucerne, if you visit the Lion Monument, on your left as you look at the monument is something called "Glacier Garden", which, if it's open in Oct., is great for kids. There is a mirror maze in there that all of my kids loved.
I've been in the Alps with each of my kids when they were young teens, and they loved the hikes and seeing cows and goats around.
As for what to do in the area, yes, you can do the Jungfraujoch trip, but many many days the top of the mountain is in the clouds or might be once you get up there. I'd recommend instead to do the wonderful trails, all easy, but fantastic.
If you have a few days near Lauterbrunnen, here are the 3 "classic" hikes in the area, and they follow the 3 great ridges.
The first goes from Grutschalp to Murren. From Murren, you can go down to Gimmelwald or you
can hike uphill on a diagonal from the Murren cablecar station, and eventually end up at the restaurant at
Spielbodalp. There you can get a drink and sit at a picnic table and watch the
cows. You can also get to Spielbodenalp via the Sprutz Falls route. About a third of the way up the trail branches near a hut. Take the upper one, you go to Spielbodenalp. The lower one goes fairly level into the woods and eventually to Sprutz Falls, which you cross under, climb up the trail on the opposite side, and end up in Spielbodenalp. You can see that trail and others here:
http://www.gimmelwald.com/pics/gimmelwald/hikemap.jpg

The second goes from Mannlichen to Kleine Scheidegg. The walk itself is
about 2 hours, and we save this for the sunniest day, because of the panoramic
views. (Leave in the morning--most afternoons the peaks cloud up!) Take the
train from Lauterbrunnen to Wengen, then a 5 min walk to the cable car
station, where you take the cable car up to Mannlichen. Once up, you can look
left at a 30 min up and back trail to a lookout point. If you look right
there is the main trail that leads toward the Jungfrau mountain, which takes
1.5 hrs and you end up in Kleine Scheidegg, where you can catch a train down
to Wengen and finally Lauterbrunnen. Great panoramic views all along the
route. This trail is so easy you could do it in a wheelchair.
It takes about 20 min train from Lauterbrunnen to Wengen, another 20 min cable car to Mannlichen, 1 and 1/2 hr hike to Kleine Scheidegg, 40 min train down to Lauterbrunnen.
The 3rd classic trail is from First to Bachalpsee. Take the Lauterbrunnen
train down to Zweilutschinen. Get off to switch to the train going up to
Grindelwald. Once in Grindelwald, walk about 20 min further into town to get
to the cablecar station, where you take it all the way up to First. From there
walk to Bachalpsee. There are trail signs, and it is about 45 min each
way. On a sunny day, you can get great pictures of the mountains reflected in
Bachalpsee, which is a small lake. Just Google image "Bachalpsee" and you'll
see what I mean.
Here's a map of the whole area:
http://ontheworldmap.com/switzerland/ski/jungfrau/jungfrau-summer-map.jpg

As the others have said Zermatt is out of the way and not worth the side trip.

In Luzern there's also the Swiss transportation museum. It's a fun interactive museum for kids. It's perfect if you run into a rainy day. Your 12 year old would probably enjoy it. Your hotel should give you a transit pass so you can ride the bus there and back. If it's nice and you're feeling energetic you can go to Mt Pilatus. It's only worthwhile if it's nice out.

As mentioned no need to stay in Interlaken. If you stay in Murren it makes going to the Jungfrau more difficult as it's on the opposite side of the valley. But it does make going to the Schilthorn easier. On a nice day the view from the Schilthorn is spectacular. As I understand it, the Jungfrau is pretty much a full day excursion.and fairly expensive even for Switzerland.

Thanks Tim, Harold, Shoe and Rocket for your advice!

Hi Tim,
Thanks for your reply! Sorry, is my mistake. I mean I have 16 days and 15 nights for this trip.

Hi Harold,
Thanks for excellent advice!! Glad that you tell me about not to stay in Interlaken as I did not know it consists of casino, a sex shop, lots of jewelry shopping, large hotels, and lots of tour groups. I will definitely follow your advise to stay in Lauterbrunnen. Do you know about any good location and clean 3 star hotel in Lauterbrunnen and other place that I plan to go? I will also just plan for the shopping trip on the London.

Hi Shoe,
Thanks for your excellent advice and the 3 classic hikes!! All 3 sound so interesting and exciting and beautiful!!

Should i buy half price Swiss pass or Standard Swiss pass?

I read some blogs, it mentions that in October, some cable cars will stop operation. Is it true?

Hi Rocket,
Thanks for your advice! I will definitely visit Swiss Transport museum! I think my girl will enjoy it!
Something new for me from you! I will go read more about Mt Pilatus and Schilthorn!

I plan to go Zermatt is because I read people posted about Matterhorn Glacier Paradise. That is why I am thinking is it worth to make a trip down...

Thanks!

Only you can say if the trip to Zermatt is worth it.

Google has some images of the Schilthorn. To me it's just as nice as Zermatt. I would post a link but I can't seem to get it too work. BTW if you go to the Schilthorn do stop by the Bond world exhibit. It has fun things like a simulated luge ride that your daughter may like.

Here's a RS video you may want to watch.

https://www.ricksteves.com/watch-read-listen/video/tv-show/best-of-the-alps

You may want to get some guidebooks for your trip. I don't know what is available in Singapore but I like the Rick Steves guidebooks. It's available from Amazon and iTunes Store in electronic form. Amazon has the physical book too.

I just noticed that your trip is mid October. I think that's shoulder season. Summer hiking season is over and winter ski season hasn't started. You may want to ask in the Switzerland forum to see what's open and what's not. I'm sure Chris F and others will be able to help. They may also offer more insight as to Zermatt.

Have a great trip.

Thanks Rocket!

I will start a new topic in the Switzerland forum to ask them only on my Switzerland trip and also what is open and close in October month.

Hope it is still not too late for me to book my hotels and Swiss pass etc. ☺️

Thanks you so much for your help!

Sorry another question.

Where can I buy the train ticket from Switzerland to Paris and Paris to London? Is there any train pass I can buy that can allow me to use in Switzerland, Paris and also London?

Thanks!

"Where can I buy the train ticket from Switzerland to Paris and Paris to London? Is there any train pass I can buy that can allow me to use in Switzerland, Paris and also London?"

You only need the pass for Switzerland. Any pass that would cover the other trips will be a huge waste of money.

For Switzerland to Paris, book on Swiss Rail: https://www.sbb.ch/en/home.html

For Paris to London, book on Eurostar: http://www.eurostar.com/

For both of these tickets, the prices go up the closer you get to travel. However, the cheapest tickets are non-refundable and non-exchangeable. Make sure you understand the restrictions, and know your travel dates, before buying.