I know that parchment at least was in use, as well as engravings in stone obviously, but what other materials to write on did they use? They must have been familiar with papyrus: was it also used in the Roman Empire? And which materials were most common/cheapest? How available were they to rich people and to common people?
By far the most common medium for everyday writing, across the empire, was the wax tablet - wooden panels, which could be doubled or formed into what we'd recognise as a book shape, with wax melted onto them. You scratched what you wanted to write into the wax with a stylus, and could melt it and pour over fresh wax when you were done with it and wanted to write something else. They seem to have been an absolutely everyday part of life for literary or business-minded Romans - you find endless references to them in the work of a Catullus or a Cicero, but also find them in the portraits of Pompeiian bakers, such as Terentius Neo and his wife. Several of those that have been found were also written with schoolroom exercises, testifying to their use in education.
Papyrus was also used - very commonly in Egypt, as you might expect, where we find it with all sorts of casual notes on it as well as more serious and permanent pieces of work. As far as I know, nobody has actually found any papyrus in one of the wetter Western provinces, but it's generally agreed that it was a major medium of writing there - it's just that it doesn't survive unless it has the sort of very dry conditions that you have in Egypt. We have found papyrus scrolls in Pompeii and Herculaneum, and literary sources (such as Pliny's Natural History) suggest that it was what you generally expected to pick up if you took a volume out of a library.
Parchment (which I'll use as a catch-all to include vellum) existed and Roman sources - Pliny again - do write about it, but it seems to have been less common - which makes sense, when you consider that papyrus was a lot easier to make and therefore cheaper, especially in the heavily interconnected empire of the 1st and 2nd centuries AD. However, parchment is more durable, and so we have some reasonably hefty documents surviving in it - the Codex Vaticanus of the Bible, for instance, from about AD 300, or manuscripts of the Aeneid and Iliad from the 4th and 5th centuries respectively. This isn't really the case for papyrus, which tends to survive as frustratingly partial fragments.
One of the big changes in the fifth century as part of what we call the 'end of the Roman Empire' is the localisation of trade and the collapse of the systems that encouraged and made possible regular long-distance exchange - this is where parchment comes into its own, as you can make it with materials that just about everyone has on hand. As such, we have a lot of copies of Roman documents on parchment, such as the Tabula Peutingeriana, a 13th-century copy of a (vaguely and complicatedly) imperial-period Roman map. There's no real way of knowing if the originals of these were on parchment as well, or whether they were on papyrus.
