A typical journey will require three changes
(1) Local train from Osaka to Shin Osaka (the Shinkansen station)
(2) Fast Shinkansen train (Hikari or Nozomi) to, Nagoya or Shin Yokohama
(3) Slower Shinkansen train (Kodama) to Odawara
(4) Local train to Hakone

You could eliminate one change by timing your departure to take a train that goes directly from Shin Osaka to Odawara. There are about 20 of those per day from Shin Osaka. (Most are slow, Kodama trains. However, there are also some Hikari trains stop at Odawara - one every couple of hours).

Search Hyperdia for trips and click the "train interval" link to see a full day train timetable for a particular route (e.g. Shin Osaka to Odawara). That will help you find a departure time that lets you reach your destination with fewer changes.