HOW TO REACH HAKONE?
There are many options to this region. Here are a couple we recommend:
Odakyu Train - From Shinjuku (Tokyo) to Odawara station (Hakone) via the Fuji Hakone Pass or the Hakone City pass (includes round trip to/from Tokyo to Odawara and unlimited transport in the region)
JR Tokaido Shinkansen from Tokyo Station to Odawara Station
Shinkansen train to Odawara or Mishima (if you want to start at Fuji Five Lakes) and renting a car to explore from Fuji Five Lakes
Also the Ltd Express "Romance Car" (partially covered by the two passes above but additional ~1200 Yen). You have to buy these in advance online or from Odakyu Sightseeing Service Center or an Odakyu ticket machine.
HOW TO SEE MT FUJI FROM THE TRAIN?
From Tokyo to Hakone you will see Mt Fuji on the right side so seats C & D of the train. On a clear day you will see Mt Fuji between Shin Matsuda Station and Kayama Station. You can also go via the Romantic express (+1200 yen) where you can book observation decks seats! If you take this train to Hakoneyu, from their you can take a tram to Gora station to explore Hakone.
TIPS
We did the last one so direct Shinkansen to Mishima and rented a car for three days! so you can do Day 1 and 3 Fuji Five Lakes and Day 2 Hakone too. (You need an international driving permit to drive here and we suggest taking complete insurance).
If you don’t prefer driving, you can also reach Mishima/ Odawara and use the Fuji Hakone Pass to experience a very similar trip
Use google maps for the best routes to travel between places, the Fuji Hakone Pass already covers most routes