Hakone has always been a favorite “getaway” spot and normally all year round.
Its proximity to Tokyo makes it a prime location for anyone wanting to go on an all day hiking trip or spending a few nights to relax amongst the beautiful mountain and natural scenery.
My advice is to make sure you reserve a seat on the Romance Car, leaving from Shinjuku. You can also purchase a 1 or 3 day Hakone Free Pass ticket from the Odakyu travel center. This includes various discounts and free rides which may save you from lining up in a queue.
The Open Air Museum is a personal favorite, followed by the cable car ride from Sounzan to Togendai. The site on a beautiful clear day is amazing. Don’t forget to have the black Onsen Egg as well. A local favorite.

Fiji-Hakone-Izu National Park is well known to many people for its concentration of so many activities to cater to any tourists need. It is easily accessible from central Tokyo, taking under one hour to get there.
Once you arrive at Odawara Station if you are coming on the Shinkansen or if using the train from Shinjuku Station, arriving at Hakone-Yumoto the area is very user-friendly for international travelers.
There is a great selection of souvenirs shops next to Hakone-Yumoto Station that offer traditional crafts or food of the area. You are also able to catch buses from this point that will give you access to the other tourist spots or your accommodation.
One of the best views is at Owakuni-dani, which is the top point of the geothermal valley. You can get here by local buses, catching the Hakone Ropeway, or driving. One of the famous things to do is eat the hot spring eggs, which are black in color from the thermal cooking process. There is also a wonderful museum that costs only 100yen, it displays historical and interactive models that teaches you about this area. Finally, depending on the weather this has a great view of Mt. Fuji and Lake Ashi.
From this point you can catch the ropeway at Owakuni Station to go in two directions. Heading back to central Hakone you would take the ropeway to arrive at Sounzan Station and from here are a number of bus options to take you to The Hakone Open-air Museum or Okada Museum of Art. The other direction towards Lake Ashi takes you to Tokaido Hakone Ropeway Cruise Interchange Station where you can catch the pirate ship and enjoy the sites dotted along the lake.
The National Park offer a lot of adventure activities of walking and hiking and these can be arranged by the getting the area brochure or asking your accommodation front desk to contact the many companies operating in this area.
This year I stayed at Bar Hotel Hakone Kazen which is about 400 meters from Okada Museum of Art. Has to be one of the best experiences I have had in Japan. The time I visited perfectly coordinated with a day of snow, this coated the area with a beautiful blanket of white.
I think you can easily spend a week in this area to be able to see all that it has to offer. I would also recommend renting a car for part of your stay if you are able to drive, this will allow you to use your time better and give you more freedom to see other less known spots.

This National Park covers a lot of different areas, which I visited on a few different occasions.

The Izu Peninsula is gorgeous. Many people think you have to go as far as Okinawa to enjoy beautiful crystal clear blue waters and tropical weather, but it's actually found right here in the Izu Islands.

When I went, I took an overnight boat to Kozushima and then camped on the beachside campground. We rented bikes and biked around the island, enjoyed snorkeling, hiking, and BBQs on the beach. It's such a fun thing to do as the summers are longer than back in Tokyo (yet, not far away!).

I also love Hakone so much, I've gone quite a few times and especially love it as a nice weekend getaway from Tokyo. I usually get the package deal with the all-you-can-ride bus/train pass which includes the boat on the lake as well.

If the weather is good, I highly recommend the Open Air Museum - it's one of my favorite museums in the world! And the volcanic area cable cars, too! If it's possible, go during off-season because it can get really crowded otherwise.