JR pass / hakone free pass

hi, I will be in tokyo from 18/3 to 20th. Then plan to travel to hakone from 21/3 and 22/3. From here i plan to go to Osaka, Hiroshima and Kyoto ie From 23/3 to 28/3 . I then return to tokyo on the 29th.
Should i get a 14 day JRpass and the Hakone free pass or will i be better off with a 7 day JR pass and the Hakone free pass. Can i travel with the hakone free pass from Tokyo to Hakone?
Thank you in advance for you advise.
Sash
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There are different versions of the Hakone Pass. There is a pass that takes over from Odawara Station, and give you access to the transportation that is not covered by a JR Pass, and there is a second that also includes transport from Shinjuku Station to Odawara Station. There is also a two or three day option on the pass.

On the 23rd, you will use the pass to get from your hotel in Hakone to Odawara Station, and from there use your JR Pass., or, you will have returned to the Tokyo area, and will activate the JR pass from there.

The cost of the Tokyo extension for the Hakone Pass is 1500 yen, and the cost of a one way trip on the Odakyu private rail line is either 880 or 1770 yen, depending on if you take the regular or the Limited Express train.

It is possible to just pay 880 for the liner train, and then take off from Odawra Station for Kyoto with your JR pass. This means that you will need the lower cost Hakone pass. You may only stay in Hakone one night, so you might leave for Hakone on the 22nd, and get a 2 day pass, which is 500 yen less. These are choices that you will have to decide.

As for the JR Pass, just get the 7 day version. The 14 day pass works out to over 3000 yen/day. The cost of a train ride in Tokyo from Shinjuku to Ueno (a possible trip on the Yamanote line) is 200 yen. This means to break even, you will need to make 15 such trip each of the 7 other days that the pass is active. Just get an IC card for the things not covered by the JR or Hakone Passes.

Which area in Tokyo will you stay until the morning of 23 March?

The cost of the Tokyo extension for the Hakone Pass is 1500 yen, and the cost of a one way trip on the Odakyu private rail line is either 880 or 1770 yen, depending on if you take the regular or the Limited Express train.

The extra cost on a roundtrip from Shinjuku is 1140 yen, not 1500 yen.
A Hakone Freepass does not cover a Limited Express surcharge.
The one-way Odakyu basic fare to travel from/to Shinjuku to/from Odawara is 880 yen.
So, unless you make a roundtrip, the Shinjuku version of a Hakone Freepass does not pay off.

However, because validity of a Hakone Freepass is for either 2 or 3 consecutive calendar days, your Hakone Freepass will expire before you will travel back from Odawara to a station in Tokyo, which you plan on 29 March.
Therefore, if you purchase a Hakone Freepass which can be used on Odakyu Lines, you will have to abandon your right to have a return ride from Odawara with the pass.

Can I travel with the Hakone Free Pass from Tokyo to Hakone?

For the above-mentioned reason, I recommend you would pay regular fees to get to Odawara and buy there a Hakone Freepass which can be used only inside the free area.
http://www.japan-guide.com/e/e2358_008.html
http://www.odakyu.jp/english/center/#tabitem-02

A 14-day Japan Rail Pass does not seem to suit your itinerary.

From here I plan to go to Osaka, Hiroshima and Kyoto

Will you travel westward from Odawara on 23 March?
If so, a 7-day Japan Rail Pass can save your JR fees from 23 through 29 March.

A Japan Rail Pass traveler is not allowed to take a Nozomi or Mizuho train on Shinkansen.
You will probably change Shinkansen trains on your way from Hiroshima via Shin-Osaka to Kyoto because a through non-Nozomi, non-Mizuho train for Kyoto will not depart except in the early morning.

Please note that Shinkansen is not available to or from Osaka Station.
Shin-Osaka Station and Osaka Station are connected by a JR conventional (= non-Shinkansen) line.

JR companies will revise train timetables on 26 March 2016.
In this revision, Shinkansen Nozomi trains will be rescheduled, while Shinkansen Mizuho, Sakura and Hikari trains will probably remain (mostly) unchanged.

Enjoy your trips!

Omotenashi is correct, the Limited Express train that I pulled up on Hyperdia appears to be the "Romance Train", and the reserved seat surcharge is not covered by the Hakone Free Pass. My bad.

I believe that all three responses agree that a 7 day JR pass will work for what you want to do. The question then becomes how long do you wish to stay in the Hakone area.

If it were me, I would go there on the 21st, and get my stuff to the hotel. I would probably do what I could of the "Round Course" on the 22nd, and stay a second night. I would then leave to Kyoto, or wherever your next stop is on the 23rd. This means that I would want the three day version of the pass.

This is up to you.

Thank you all. I plan to travel from Tokyo to Hakone on the 21 st. I will be staying in Hakone on 22nd as well. Want to depart Hakone for Osaka on the 23rd. On the 24th I will depart Osaka for Hiroshima where I will stay till the 26th. On the 26 th I will depart for kyoto and stay here till the 28th. On the 29th I will depart Kyoto to go back to Tokyo..
I understand that from the discussion I will have to activate my 7 day jr pass when I leave for Osaka. Is this correct.
Do I purchase the Hakone free pass to get me from Tokyo to Hakone ? Is this correct? Thanks you all once again.