None of the forms of magic are actually proprietary to the city their guild is based in. Each city has their own casters of many different stripes. The Guilds perform specific function in their city-state. Thaumaturges, among other things (*cough* assassins *cough*) the priesthood of Ul'dah's whacky money religion, Conjureror's help Gridania communicate with the Elementals, Arcanist serve as customs agentes for Limsa, and Astrologians read the stars to predict dragon attacks for Ishgard. While people are encouraged to serve the guild and city-state in exchange for instruction, it is not required. It also is not necessarily required to have learned a character study magic at one of the guilds (Yssen didn't, he learned his Thaumaturgy and what not from his terrifying ex's mom). Similar smaller/master apprentice methods of instruction should be perfectly acceptable. Heck, if you want connections to other PCs, ask for someone to play your character's mentor (or see if they mind sharing a mentor).
Long story short, ya have options as far as the basic schools of casting go. Yar.
EDIT - Slight lore tid bit worth mentioning. Pureblood Garleans can't cast magic. Imperial casters are either conscripts from lands they have taken over, or using magitek devices to emulate magic. That said, they have a LOT of conscripts and magitek. Yar.
Long story short, ya have options as far as the basic schools of casting go. Yar.
EDIT - Slight lore tid bit worth mentioning. Pureblood Garleans can't cast magic. Imperial casters are either conscripts from lands they have taken over, or using magitek devices to emulate magic. That said, they have a LOT of conscripts and magitek. Yar.