Sherry,
Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you probably won’t find statutory birth, death or marriage certificates in Ireland for this family. For earlier years you usually need to rely on church records, where they exist. You obviously need to know the precise denomination in order to search the correct records. Not all churches have records for that period and not all are on-line.
RC records are mostly on-line on the nli site:
https://www.nli.ie/en/family-history-introduction.aspx
For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
The 1831 census for Co Londonderry has mostly survived and is on-line on this site: http://www.census.nationalarchives.ie/search/
You can search it to see where the various surnames were found i9n 1831. Expect the spelling to vary. People didn’t bother about standard or accurate spelling then and so you will find many variants. Grue for example is Grew. Starrett can be Starat, Staret etc. Search under Star* and you should all the different spellings.
But if you have no idea where in the county the Gamble or Beattie families lived then you would just have to work through all the parishes in the county. Starting perhaps with those where Grew is found (3 parishes) and Starrett (5 or 6 parishes). Most of the records you need will not be on-line. You need to go to PRONI to search them.
Elwyn
