I think it's rather fun to get to Hakyone Yumoto from the Hyatt. If you have significant baggage, take a taxi from the hotel to Gora. Otherwise, the cog railway is about a 5 minute walk and it will take you down to Gora (although it's not that frequent, check the schedule).
From Gora, the quaint Hakone Tozan railway takes you to Hakone Yumoto, running every 15 - 20 minutes, although it takes longer than a bus or shuttle. I particularly liked the three reversals (where the engineer has to switch ends and the train switches direction) to get around steep curves.
From Hakone Yumoto to Odawara, there are regular trains that connect to the Hakone Tozan line, not just Romance Cars. And no, the Romance Car is not very romantic, but it is quite comfortable. Note that although the Romance Car originates in Hakone Yumoto, it also stops in Odawara.
Odakyu runs a number of limited express trains on the Odawara - Shinjuku route, not just Romance Cars. They are all comfortable.
There are also plenty of regular rapid trains on the Odakyu line which connect from Odawara to Shinjuku, if you want to save money. Odakyu is less hassle than taking JR to Tokyo and transfering to Shinjuku. Also cheaper.
I suppose you know this already, but the JR pass is not valid on Odakyu line trains. So if you have a valid JR pass, take the JR route from Odawara. However, if you have a JR pass, go ahead and do silly things. For example, on my last trip using my JR East Pass on a valid day, I took the shinkansen from Ueno to Tokyo just because I could.