Hi Tristan!
For the travel you list the only rail pass that covers the JR zones you will travel through (JR-East, JR-Central & JR-West) is the country wide Japan Rail Pass (or JR Pass). This rail pass covers all Japan, and is only marginally more expensive than single zone passes.
A 14 day rail pass will cover you from the 25th July to 7th Aug, which is the period which covers the majority of your long distance travel, however will cost ¥45,100. Your ticket cost will have to exceed this for a 14 day pass to be worthwhile, so lets take a look at your actual JR costs:
- Osaka > Tokyo: ¥13750, (194mins, 556km)
- Tokyo > Kyoto: ¥13220, (164mins, 513km)
- Kyoto > Tokyo: ¥13220, (161mins, 513km)
TOTAL: ¥40190 for 3 legs
As you can see, based on your current itinerary you will not make savings with a 14 day pass over individual tickets. This means that unless you plan on making more JR trips, I recommend that you purchase individual bullet train tickets as you need. If you do more JR travel during this time, you only need another ¥4910 of JR travel for a 14 day rail pass worthwhile and for any further JR travel to be free, but this will very much depend on your itinerary.
I'm afraid that as the JR Pass is a discount ticket there are no further discounts given to under 25s on the nationwide pass.
For the subway, there is no pass that will cover the entire period across both cities subways, however you can purchase prepaid cards that will make travel more convenient. You can also purchase commuter tickets for each city if you think you will be travelling a lot within the cities. Here are pages on the travel cards/passes for travel in [Osaka] and Tokyo that you should find useful.
Hope this helps - if you have any further questions, please ask!