1st family trip to Japan

We're a family of 4 (2 young teenagers) on 1st trip to Japan arriving at Narita on Dec 19, 2016 and departing on Jan 1, 2017.
Hoping to do some skiing, pick strawberry and visit Hakone to view Mt. Fuji. Planning to base our stay at both Ichikawa and Karuizawa.
Please advise for the above itinerary like best place and chance to ski as well as suggest places to visit.
We like nature and planning to skip going to Universal and Disney.

Strawberries are not winter fruits.

I'd suggest doing some more of your own research first; it's not fair to expect a forum to plan your holiday for you. You've said very little about your trip: where you want to go, what kind of travellers you are (fast/slow), what your budget is, how long you want to ski for, etc. If you don't want to do research and plan the holiday yourself, you should visit a travel agent.

You should get started ASAP to prevent things becoming booked out!

I've done some research but am asking for some suggestions and ideas which can help me to decide. Thanks.

In that case, present the research you've done so no one repeats something you already know. If will also give people a better idea of what to suggest, because it will illustrate what you like regarding travel, and give an idea as to your budget.

The more information you provide, the better responses you'll receive.

If I was you and your kids were my kids when they were young teens I would take them to Akihabara for the anime/otaku/video game opportunities (its easy to censor where they go as there are tons of arcades and you can go in and check). The "bad" stuff is less offensive than the cartoons kids watch at home. Then, as hero dad, I would select either Universal or Disney if the budget allows it. Otherwise there are themed resorts in the Izu Peninsula that might work and save some yen. If you're really adventurous, there is a Ninja museum/show at Iga-Ueno east of Nara but you would need a day to get there, watch the very good martial arts show, then get back to the shinkansen. Finally, I would look around this site to find a less trendy ski resort than Nagano (there are tons) and have a fun family ski-day before going home.

1) Regarding skiing - thinking of either doing it at Gala Yuzawa at Niigata or Prince Snow Resort at Karuizawa - wondering which is better during our period of stay in Japan?
2) Thinking of going to Hakone to enjoy a good view of Mt. Fuji - can anyone suggest where is a good place to stay for 1 night? How is Kawaguchiko Hotel and Oshino Onsen? Or is a day-trip by driving from Ichikawa, Chiba good enough?
3) There is a Harada Farm at Numata where they grow strawberry in greenhouses - so can pick strawberries there. Still trying to check out other places.
4) When in Karuizawa - we'll visit the Snow Monkey Park.

Overall, we enjoy nature and going to good parks but also want to see some aspects of the Japanese culture - going to Shibuya, Harajuku, Shinjuku and Tsukiji market. We hope to try their excellent food as well.

However, we'll skip going to Universal and Disney as "been there, done that" somewhere else.

Thank you / and johnnyman for your ideas and giving me your time in writing them.

Odaiba
Akihabara
Harajuku
Shibuya
Shinjuku
Ikebukuro
Railway Museum, http://www.railway-museum.jp/en/
Nikko Edomura, http://edowonderland.net/en/

Karuizawa is not a good base, I think. There should be little snow. Their ski resorts are covered mainly by snow machines. Basically, you need to go to the Japan sea side, the north or west side of mountains. Karuzawa is the south of Mount Shirane...

Yuzawa Onsen, Nozawa Onsen, Shiga Kogen, and Hakuba are the popular sites. In Dec, Zao in Yamagata, or Appi in Iwate should be more promising.

If you mean Jigokudani Snow Monkey Park, here are some access details: http://www.snowmonkeyresorts.com/access/

I don't see Karuizawa as a good access spot on that website.

Skiing, onsens, shopping. Being the start of winter, you have limited options compared to warmer months.

The above response starts of on the basics with some indoor options also.