The period that you're referring to is often termed "Late Antiquity". It can be described roughly as comprising the period in which the Roman world became Christian and then in large part ceased to be Roman, and in which the East became Muslim, leading to more recognizably "medieval" state systems as broadly defined as that might me. In other words, from about 200 AD to 700 AD, perhaps with an extra hundred years on either side of that tail end to provide context (or, at the later end, to consider 8th century developments in Islamdom that would become themes for the entire Medieval period.)

The man often credited with founding Late Antique studies is Peter Brown, and his book The World of Late Antiquity remains as good an introduction to the topic as any, though there are of course many other such books since then. It has the advantage of being both short and highly readable.

Glen Bowersock is the other big name in Late Antique studies. They also co-edited a (somewhat unusual book) along with Oleg Grabbar called Late Antiquity: A Guide to the Post-Classical World which is part essay collection, part encyclopaedia. The essays are well worth while if you can find a copy of it.

I can point you to other worthwhile reading on Islamdom, but perhaps one of our Medievalists/Byzantinists could chime in on Europe.

edit: typo