I am trying to find out what ship my father served on during WW2 , as when he was alive he never spoke about the war. I believe he left the Navy as a Petty Officer and served in and around Burma. I do know that he got shot in the leg. His name is Graham Charles Jones from Swansea, South Wales.
As @TomH indirectly suggests, it is possible to request your father's Service Records from the UK's Ministry of Defence. See https://www.gov.uk/requests-for-personal-data-and-service-records, in particular the section "Service records of deceased Service personnel".
Obtaining Service Records has a bad reputation because of the time it takes and the necessary steps along the way, many of which act as blocks to many applicants. However, assuming you have your father's death certificate and that you either are his next of kin, or can obtain consent from his next of kin, then yours should be a simple (if lengthy) route to his full service history. I know of no other guaranteed way to this data.
An administration fee is payable (Freedom of Information is not Free Information).