You start at I materia and as far as I can see at the moment it will require multiple I materia before you unlock the "II" materia meld. And so forth to IV. What I don't know yet is whether or not you'll still be able to meld the I materia once the II is unlocked. The help text says that increasingly powerful materia will be needed as infusions progress. This makes it seem like in order to hit a cap for a particular cap you'll need IV's.

Alexandrite can be obtained from treasure maps. They cost 800 myth or a daily Expert Roulette quest. Might be weekly? It can also be obtained as a low drop from FATEs. One Alexandrite is needed for each meld.

Novus requires an initial 1500 myth buy in. You have to buy 3 superior something inks to turn in for the sphere (or spheres) that correspond to your weapon.