Scholar Question: How Should I meld my Astral Ring?

It depends on what stats you want from your Astral Ring. Vitality and Piety can be capped with a single Grade IV Materia, so if you want any of those stats in, it's preferable that you put those in first, unless you're gloriously rich and can afford to forbidden meld Grade IV Materia. Spell Speed (Quicktongue) and Crit rate (Savage Aim) will require at least 2 slots regardless if it's Grade III or IV, so doing two Grade III of each is the better way to go about it (and cheaper... probably).

As for the Materia slots in your gear, the visible ones are simply the amount of Materia that can be melded without fail. Accessories such as rings can only accept one Materia without fail (called Forbidden Melding). Beyond that, there's a chance that the Materia will fail to be inserted into the piece of equipment, with the Materia being destroyed as a consequence. The failure rate increases with the Grade of the materia and the number of materia being inserted, thus it's cheaper to insert the most expensive materia first. As an example, a Grade III Materia will only have a 35% success rate at the first forbidden meld, dropping down to ~22% for the second forbidden meld and so on.

Anyway, some example melds, I suppose. If you just want to maximize Crit and don't care about anything else...

First Slot: Savage Aim IV (or Savage Aim III)
Second Slot: Savage Aim II (or a second Savage Aim III)
(Third Slot: ...or you could just add Savage Aim II in all the slots.)

Want to maximize some Vitality or Piety?

First slot: Vitality IV/Piety IV
Second Slot: Savage Aim III
Third Slot: Savage Aim III

Got Gil to burn and want all the stats? Keep in mind, the three first slots can switch spots, with the most expensive materia being melded earlier.

First Slot: Vitality IV
Second Slot: Piety IV
Third Slot: Quicktongue IV
Fourth Slot: Savage Aim III
Fifth Slot: Savage Aim III