I've actually researched quite a bit into the symbology (

) of the Zelda games;
Kokiri's Emerald resembles/is based of the Kokiri/Deku symbol (as seen on the Deku Shield)

Goron's Ruby in turn has similarities with the Goron Footprint

...And Zora's Sapphire is similar to the symbol above King Zora's head (which I was not able to get a pic of) and also the centrepiece of JabuJabu's 'crown'

The Kokiri symbol can also be seen in WW, on the doors of the island opposite Forest Haven, and I believe also on the doors of the Forest Temple in
TP.
The Goron symbol is found on every goron's arm (except Link-goro I think), albeit with wings in MM, although I can't find a pic of it. If memory serves, the symbol is also on the doors within the Goron Mines.
The Zora symbol, as you know, is also similar to the symbol of Nayru's Pearl (WW) & the Tune of Ages (
OoA) - except it is inversed (I think).
Finally, and as a personal theory, if you look at the pictures above of the spiritual stones, each one has a number of golden 'spikes'; the Kokiri's Emerald has one at the bottom, the Ruby has two (top-left & top-right), while the Sapphire has three at equidistant points around the stone. This leads me to believe that the stones were designed to be in that order. (ignoring that this is just a game, where everything was designed by Nintendo)
Now, with the medallions, not much can be said, so I'll try to be brief;
The Forest Medallion is similar to the Bombos Medallion, and has four things which I believe represent leaves.
The Fire Medallion represents a fire. (I can't find much on this)
I believe that the Water Medallion represents ice (not much here either)
The Spirit Medallion represents Yin & Yang; it is similar to Din's Pearl (as mentioned earlier) and one of the tunes in
OoA (The Tune of Echoes I think; I'll verify it later). It is also similar to the Quake Medallion
The Shadow Medallion and Light Medallion are opposites; they are based on triangles and circles, shapes known to be physically strong.
Also, Farore's Pearl is shaped the same as the first tune in
OoA (I forget the name)
There's more, but that's all I can think of at the moment.