As a Scandinavian I know a fair bit about the Viking Age in general and I'd say I'm fairly well versed in the subject for a layman. However, I'm currently planning a long vacation in Britain (Scotland and England) and I'd really like to delve deeper into the history of the Norse conquest of the British Isles. I'm specifically interested in the Norwegian conquests. Any good introduction to the subject would be much appreciated, I don't shy away from more scholarly works.
As a follow-up I'd also appreciate a recommendation on a good introduction/overview to Anglo-Saxon Britain. I think it'd be interesting to get the other side of the story, the culture that interacted with the Vikings.
If you're into archaeology and the societal impact of norse migration to Britain I highly recommend picking up two books by Francis Pryor:
Both are well written and easy to read introductions to both anglo-saxon and viking age society in Britain. Britain AD was made into a television series as well, although personally I much prefer the book.
The Norman conquest: The Battle of Hastings and the Fall of Anglo-Saxon England by Marc Morris is a pretty good resource to start out with. It does well at breaking down the history of Britain and the Scandinavian influence prior to 1066 and the factors that led to William's conquest.