The cheap way to travel between cities

Please would like to know the cheapest way to travel between tokyo/hakone/kyoto/hiroshima/osaka in 2 weeks ?
arrived in tokyo /departure from osaka (family with 2 kids 12 and 15 years).
is this itinerary perfect ?And how many days in each city ?we are interested in nature and culture .food and people ...
Thanks a lot

Itinerary is fine.

Cheapest way to travel - considering the cost of flights, accommodation, food and sightseeing, why does everyone then want to be cheap on internal transport - there is practical and then there is cheap.

OK, cheap - walk, bike, but clearly not practical. Highway bus is more expensive, but time consuming and about the same price as local trains. But you lose many DAYS of your holiday wasted travelling slowly. The other consideration is not to do unnecessary things. For example, in Hakone the "loop" covered by the Hakone Free Pass includes the ropeway and pirate ship. Just using the buses to get around the area is cheaper, but coming to Japan and not doing anything defeats the purpose of your trip. (You might blow your travel budget for the day by visiting the Open Air Art museum instead.)

Practical, get the shinkansen/bullet train. It's not actually that expensive given the time saving and the distances covered and also as a proportion of your overall travel cost, it's not that much.

For Tokyo to Hiroshima and back to Osaka the 14-day rail pass is not the best option, single tickets are better.

Difficult to know without seeing your itinerary. Unless you can fit in from the day you leave Tokyo to the day arriving in Osaka all within 7 days then the JR 7 day nationwide pass will pay off. Use IC card while in Tokyo and Osaka.

Alternatively may consider JR Kansai Hiroshima Area Pass https://www.japan-guide.com/e/e2361_kansai_hiroshima.html for travelling between Kyoto - Hiroshima - Osaka. However Shinkansen between Kyoto and Shin-Osaka not allowed. Only other JR trains are allowed.

Recently I planned my son's trip to Europe including Belgium so I know there are many good passes for travelers which is very cheap. So I guess the OP (from Belgium) might think there might be some kind of cheap (and convenient) option for travel.

Japan is in general cheap as compared to Europe (accommodation and eating out) but the exception is transportation. Paris to Brussels and Brussels to Amsterdam is only 29 Euro but in Japan the fare for the same distance is twice or even three-times more expensive.

I think the practical way is to buy Japan Rail Pass 7 day (29110 yen) and use it for your long distance travel (Tokyo to Odawara to Hiroshima to Osaka/Kyoto) which is somewhat cheaper than the normal fare. Tokyo and Kansai (Kyoto, Osaka etc.) are in both ends of your itinerary where you can travel with IC card (no discount but just for convenience).

The way the OP is written it looks like Tokyo to Hiroshima and exiting via Osaka over two weeks, so the 7-day pass doesn't cover the days and the 14 day rail pass is too expensive.
I did remember last night that there is a nationwide bus-pass that is available - spotted it one day earlier this year on the web - that might be an option, but while it might be cheap in yen, it is still expensive in time (people simply don't value their time properly, especially on a time limited exercise like a holiday).

Shinkansen Tokyo to Kyoto 14,100ish
(a little cheap for non reserved)

5 day hiroshima kansai pass 13500

Total 27,600 yen

IC card everything else.

The cheapest way to travel is by train.

Get a JR rail pass.

you will save lots of time and money!

For more info visit https://www.go-a2b.com/

Don't overlook flying. You can fly Tokyo to Hiroshima for as little as 8000 yen if you book early. And it's only 90 minutes plus ground transfers. Then the only other expensive leg you have to cover is Hiroshima to Osaka on the Shinkansen for 10000 yen. Beware of luggage fees on the airplane if you are carrying a lot, however.