Cumberland is still served by the Cumbria Archive Service and the principal for the county is Carlisle Record office
but there is much material for the West of the County at Whitehaven Record Office
However there will also be some useful material at other record offices such as York (Borthwick Institute) and London (National Archives).
Other relevant material might be found at Barrow or Kendal or Preston
The Cumbria Archives catalogue is available online with guides for different record types.
This includes Cumberland Quarter Session records. Individual names are indexed.
The Lancashire Archives catalogue is also online, and for instance has over 34,000 catalogue references indexed as "Cumberland"
The National Archives Discovery catalogue indexes the holdings of the National Archives and 2500 other archives across the UK. There are over 51,000 catalogue references for "Westmorland" of which over 8900 are downloadable.
Genealogical material on Cumberland held in Carlisle Library (and also at Workington and Penrith) is described on the Cumbria Libraries Local Studies website.
The Society of Genealogists Library in London also has much useful material for members.
Cumberland - Genuki - Archives and Libraries - links and information.