Is there an easy way to travel from Osaka to Hakone by train? The options I have been looking at have at least 3 transfers.

The route I am thinking of doing is Tokyo-Osaka-Hakone-Tokyo.

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.

With or without the JR pass?

Any case, since there is no Hakone Station, you need to make Shinkansen connections from ShinOsaka to Odawara Station.

From Odawara it is other trains buses, tramways etc go into see Hakone.

Yes, once you arrive at Odawara, you should buy the Hakone Free Pass, which lets you take the whole set of sometimes hokey and touristy means of transportation through the area--train, bus, funicular, aerial tramway, even fake pirate ship. There's a standard route using all these means of transportation.

And from Hakone to Tokyo, there is the Odakyu line limited express from Hakone-Yumoto to Shinjuku station.

There is a Hikari shinkansen leaving Shin-Osaka at 8:13 and every two hours afterward (10:13, 12:13, ... ) that stops in Odawara two hours and twenty-one minutes later without transfer required.

Done this trip few years ago and took a direct Hikari from Shin-Osaka to Odawara. But it still took 4 trains to get to my ryokan.

Osaka
| Local Train
Shin-Osaka
| Hikari
Odawara
| Odakyu local train
Hakone Yumoto
| Tozan Railway
Gora (or whereever your ryokan may be)

JR Pass will save some money if the Tokyo-Osaka-Hakone-Tokyo trip fits within 7 calendar days.

Yeah, that's about right. It's not really as hard as it sounds. The connection at Hakone Yumoto, for example, requires walking about 200 feet from one train to the next: no changing levels.