Hi there!
Thinking from the title that this would be a topic discussing the merits of the Japanese cuisine, so I was a little surprised when it was a about rail travel. Nevertheless you are at the right place!
As you are aware only people that enter under a tourist visa are eligible for a JRPass, still there are discounts to be had for students and people traveling without one.
JR does offer a 20% discount on normal train tickets for student if the distance traveled is over 101km but the student will require an official certificate provided by the school or University. Every university have their own way of doing this and some don't offer this at all, so I would recommend contacting Doshisha about this. This discount is also refereed to as Gakuwari (lit. student discount).
Additionally you can buy a tickets in advance at so called "Discount ticket shop" these shops allow you to buy tickets in advance with (a small) discount.
A 3rd option is to buy return tickets, some routes allow for discounts if you buy the return tickets at the same time. This can depend on the distance and trains involved, the best thing to do is to ask the JR Staff if a return discount is available for the traveled route.
All the above is very handy if you travel on the Bullet rail or other limited express train. Local travel generally is not very expansive so you'll be find with normal tickets. This includes travel within Kansai, such as Osaka and Kobe. The Suica card also works in Kansai so you can keep using the one you have or you can opt for the Icoca + Haruka package which is the same idea as the Narita + Suica package but for Kansai. The Icoca + Haruka includes an Icoca card (which is like Suica) and a nicely discounted ticket on the Haruka Express, which brings you to Kyoto in Style and comfort.
I think this should cover all your questions so far, let me when you brew up some more :)
-Daniel
