From the time of the Republic up through the Empire, how did Rome go about spreading the Latin language to its new provinces? Were there some sort of schools for it? What types of people would need (or have the option) to learn it? Who would teach it?
Secondly, I realize the answer to this question may look very different in the Eastern provinces where Greek was already widely spoken and the "civilized" infrastructure was better. Yet, I'm still curious, what types of people would have needed to learn Latin in the East in order to do their jobs effectively under Roman administration?
Finally, did the methodology of spreading Latin education change, say from the time of Julius Ceasar until Trajan's conquests were being incorporated into the Empire?
A good source for this is The Blackwell History of the Latin Language.
On the whole, the Romans did not actively mandate the use or spread of Latin, or at least not in the same way that the English in Ireland mandated the use of English and suppressed Irish, or how Native American schools sometimes punished the use of native languages. They were also, however, mostly unwilling to learn the languages of their subjects (Greek excepted), so of necessity any Briton or Gaul who had business to carry before the local Roman governor had to use Latin, or a translator.
Latin didn't actually spread very quickly outside of Italy. We're used to the idea of the Romans conquering a province, and then that province spoke Latin, but it didn't really happen that way at all. Some parts of Gaul still spoke Celtic in the 4th and 5th centuries (according to Jerome), and Christian bishops sometimes had trouble communicating with their parishioners. St Augustine says (Epistle 17) that Punic was still in use in Africa in his day among some of the common people, and the Romans had been involved in Africa since the Punic Wars about 600 years previously. (Incidentally, Latin probably never took much hold in Africa: it vanished quite quickly during the Islamic expansion, and Arabic replaced both Latin and Punic, probably faciliated by the fact that Arabic and Punic are closely related. See "Empires of the Word" by Nicholas Ostler.)
Generally, Latin became the native language only where large numbers of Romans settled. This usually meant soldiers: the Rhine frontier Latinized more quickly than the interior parts of Gaul. The modern Romanian language is usually thought to descend in part from the slang used by Roman soldiers stationed in Dacia.
Tacitus, in the Agricola, does note that the Britons learned Latin and studied rhetoric, along with adopting baths and other features of Roman civilization. But this was probably primarily among the urban elite only and never took root among the common people: thus, when the Roman legions left Britain, almost all traces of Roman civilization and Latin vanished.
In the East, as you note, people rarely learned Latin, and even the Romans tended to use Greek as the regular method of government. Somewhat ironically, this seems to have changed only when the power of the Latin-speaking half of the empire was on the decline. Libanius, a Greek from Antioch in the late 4th century, never learned Latin, but complained that the youth of his day were studying it in order to apply themselves to the study of Latin law. Latin was never the usual means of communication in the Byzantine Empire, but was used even there for legal terminology.
Related to this, a very interesting book just came out recently, Learning Latin the Ancient Way, which contains a number of (mostly translated) examples of Greek-language manuals on learning Latin. They mostly involve law or other forms of public business, and almost exclusively involve spoken rather than written Latin (Latin words tend, for instance, to be transcribed into Greek: ρηγνουμ for regnum, μαγνα for magna).
Follow up question: How were language differences dealt with in the legions? Were speakers of different languages grouped together in legions, with their own languages, rather than Latin, used as the primary form of communication?