Going to Kawaguchiko by train from Tokyo

Hi,
I plan to go on a day trip to Kawaguchiko from Tokyo on April 18. All the day tours are booked so we will go on our own.
I read the guides and have a couple of questions:

1. Can we buy the limited express train tickets on the day we ride the train? Or should we buy ahead? (we will be arriving in Tokyo on April 17)

2. Can someone guide us where exactly in shinjuku station to buy the tickets?

TIA!!
John

1. Can we buy the limited express train tickets on the day we ride the train?

Yes, until just a few minutes before departure. In the unlikely case that reserved seats are booked out, you can ride in the non-reserved section.

Or should we buy ahead? (we will be arriving in Tokyo on April 17)

If you wish to make seat reservations and know your schedule exactly in advance, I recommend to make advance reservations.

Can someone guide us where exactly in shinjuku station to buy the tickets?

At the JR ticket counters. There are multiple of them around the station, usually just next to the ticket gates.

Note that access by direct bus is cheaper and considered easier by some people (except maybe finding the bus terminal):
http://www.japan-guide.com/e/e6905.html

Thanks Uji!

Hello Uji,
Thanks for your help. Can I ask a few more questions about this trip?
We plan to leave Shinjuku for Lake Kawaguchiko by 8am.
The spots we plan to see are:
The Music forest and Chureito Pagoda (hopefully cherry blossoms are still in bloom)
Then Oshino Hakkai
Then off to the Hakone Shrine in Lake Ashi (by bus not boat)
End the day at the Gotemba Outlet before heading back to Tokyo.

We just want to enjoy the scenery and take nice photos ( of fuji, the Torii in Ashi, lake, Sakura, Pagoda, etc)

My questions are:

1. Are these good spots to enjoy the scenery and take photos in the area?
2. Is the planned itinerary too much for one day? (Maybe we can skip Oshino)
3. What sequence do you suggest given the scattered locations and public transport?

Thanks so much!!