A good place to start is usually the UK's National Archives. These contain a large amount of information compiled by the Royal Navy during the war. Ships' logs come under the heading ADM 53, found at http://discovery.nationalarchives.gov.uk/details/r/C1762, but unfortunately, this doesn't include records for destroyers during WW2 - these didn't survive the war. You also have ADM 177, which lists which ships officers of the RN were serving with, and ADM 187, which lists where those ships were. Otherwise, searching the general Admiralty collection for battles in which the ship served can be somewhat productive. For example, this record might give some information on Redoubt's activities.

You can obtain a copy of your grandfather's service record directly from the MoD, at https://www.gov.uk/get-copy-military-service-records. However, this won't really give much more information than any medals he received, any medical issues he suffered from during his service, and the units in which he served. In your case, this might not tell you anything you didn't know.

If you want to look elsewhere, then HMS Redoubt's service history has been collated on naval-history.net, at http://www.naval-history.net/xGM-Chrono-10DD-53R-Redoubt.htm. This doesn't give any crew information, but does give comprehensive information on what she was doing at any one point during the war - I particularly like the entry for 1st February 1944, where she sustained damage after ramming a whale. In a similar vein, uboat.net gives some info here. Book-wise, Ships of the Royal Navy by Colledge and Wardlow will give a basic overview of her activities, as will any of a number of books on RN destroyers, such as Cocker's Destroyers of the Royal Navy, 1893-1981. I'm not aware of any books focussing solely on the R-class and their activities.