Since you're coming from the States, I agree with spending the first night at Shinagawa. Even if you somehow made it to Odawara, it would be dark, and you wouldn't be able to see anything anyway.

I suggest staying up as late you can (10PM will be 6AM Pacific Time), so that you accomplish the equivalent of an "all-nighter." Then you will have a better chance of sleeping a few hours straight through. Eight hours' sleep will put you at 6:00AM. You can have breakfast at your hotel and then head over to Shinagawa Station.

If you have validated your Rail Pass at NRT, simply board the next train that is labeled a "Kodama." (The Hikari and Nozomi never stop at Odawara.) The alternating Japanese/English signs tell you exactly which stops each train will make.

Go through the turnstile that is farthest to the right and show your JR Pass to the booth attendant. Don't try the mechanical turnstiles, which are for single tickets only.

When you find your train, go stand at the point on the platform where signs say "unreserved seats." One of the impressive things about the Shinkansen is that it stops so that the doors of the cars line up EXACTLY with the lines on the platform.

Pick any seat in one of the unreserved cars and sit back. An advantage of going in the morning is that you will actually see things out of your window.

Once you are settled in Odawara and want to start exploring Hakone, buy a Hakone Free Pass, which gives you access to a standard circuit of various types of conveyances, including trains, buses, an aerial tramway, a funicular, and even a fake pirate ship.