Some of the major English-language medieval history journals, off the top of my head:
Speculum - Journal of the Medieval Academy of America, publishes not only history but lots of literary and cultural studies too.
Journal of Medieval History - Very wide range of stuff published; quality is mixed.
[Early Medieval Europe](http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1468-0254) - The leading journal for the Early Middle Ages.
Journal of Medieval and Early Modern Studies - Does more than history, and more than medieval.
Crusades - Publishes research on the Crusades.
Medieval Archaeology - Does what it says on the tin.
Peritia - Journal of the Medieval Academy of Ireland, for Irish medieval history and Celtic studies.
Anglo-Saxon England - For early medieval English history.
Anglo-Saxon Studies in Archaeology and History - More archaeology focused than the above.
Viking and Medieval Scandinavia - For Viking stuff.
Networks and Neighbours - Open-access early medieval journal. Relatively new.
The Mediaeval Journal - Very new journal, out of St-Andrews, Scotland.
It should also be remembered that lots of medieval historians publish articles in non-medieval/general history journals such as Past & Present, English Historical Review, Transactions of the Royal Historical Society, American Historical Review, Journal of Ecclesiastical History, Journal of Economic History, Social History, History, Church History and many, many more.
As for subscriptions, I can't say too much. Most academic journals are accessed through university subscriptions, because individual subscriptions tend to be crazy expensive, especially if you want more than one journal. There are an increasing amount of articles being published open-access, however, and it's worth looking into repositories like JSTOR - the often let you access a certain amount of articles a month for free, and their subscriptions tend to be much better value for money than the publishers' subscriptions.