Greek and Roman theater are very different beasts, as are their productions. We have very little information about the theatrical performances made outside of Attica before around the 4th Century--we know that they existed but we know little about the details. At Athens and elsewhere dramatic performances occurred only during particular religious festivals, the most important being the City Dionysia and the Rural Dionysia. Dramatic performances, particularly the tragic productions, were not the only performances--the City Dionysia included a procession to Dionysus, including dithyrambic performances, and at Athens each of the phylae put together a chorus of 50 boys and a chorus of 50 boys each. The other three days of the City Dionysia were devoted to dramatic performances--each of the three competing tragedians produced a trilogy and accompanying satyr play each day, with the five comedians producing their single plays all on the last day. As a religious festival the Dionysia was prepared months in advance--as soon as the archons took office they appointed the chorus-leaders, who began immediately organizing the chorus for each of the plays. Rehearsing the chorus and the actors was the duty of the competing poets. Besides rehearsing there wasn't much more to prepare--stage scenery was relatively simple, and since even by the end of the 5th Century only three speaking actors were ever present on stage at a single time in a tragedy (though comedy might have many more) the vast majority of the rehearsal must have been for the chorus.

You'll notice, particularly in regards to your question regarding the reading of the texts, that these actors and chorus members were amateurs--professional actors don't start appearing until New Comedy. The professional actors of the Hellenistic and Roman Periods seem to have been able to rapidly memorize dialogue--it was their job, after all--and in any case by that time it was often the same few works being performed over and over again. That the scripts were something of a combination of written text and oral direction is clear--even in late antiquity plays did not contain written stage directions. Attic theater, particularly tragic theater, was not like our own. The key part of an Attic tragedy is the chorus and their Dionysiac hymns--originally dramatic performances consisted entirely of the chorus, and the addition of a single speaking actor late in the Archaic Period merely enabled dialogue between the actor and the chorus. These are works of lyric poetry--the lines are all in verse, and would've received musical accompaniment. Delivery of lines was therefore not intended to be lifelike, but to be poetic, something that you don't necessarily need extensive training for--later professional actors were sometimes praised for their lifelike acting, although more frequently you find them being praised for inciting an emotional response. Consider that an enormous chunk of the male citizen population of the city would have performed in the Dionysiac ritual--between the choruses, dithyrambs, and the theatrical performances you've got an awful lot of people participating in this. Estimates of literacy rates in antiquity are fraught with challenges, not least of which is the difference between fluent and "working" literacy, but this is often cited as evidence for a widespread and fairly skilled level of literacy within the Athenian state--there are other places where Athenians speak as though all citizens are expected to be able to read. That said, the memorization of poetry was quite common in Athens--even Socrates was able to rattle off (and modify at will) verses of Homer--and tragic poetry is not particularly difficult to memorize. Undoubtedly a combination of textual and oral memorization methods were employed

Roman and Hellenistic theatrical performances worked very differently. Early forms of Italian theater and of Greek theater being performed in the west are not particularly well-documented, so not all that much can be said about them. Like Attic theater, Roman theatrical performance took place during particular festivals--the first Latin adaption of a Greek play, translated by Livius Andonicus, is traditionally said to have occurred at the ludi Romani in 240. By the time the Romans established tragic and comic theater within their society, professional actors were well-established in Greek circles, and they quickly became a class in the city--by the late Republic we see famous actors like Aesopus and Roscius. We do not know how the chorus functioned or was set up in Roman theater--since most tragedies that were produced were "reruns" of Attic or early Hellenistic tragedies we should assume that there were still choruses, but we know little about how it worked (the chorus had also fallen out of comedy by the end of the period of Old Comedy, so that's that). We know from Plautus some details about how his comedies worked. The curule aediles, who organized the ludi, commissioned professional performance troupes to produce the plays performed. It appears that the leader of the troupe selected the play to be performed himself and purchased it from the poet--in Plautus the troupe leader plays the lead part and mentions commissioning a costumer and composing music for the play. But there were many types of theatrical performance besides tragedy and comedy--by the Principate mime and pantomime were much more popular, and it's unclear how these were staged and produced.