Hi David, as Brian has suggested your first step is to obtain your mother's birth certificate - as you have her exact date of birth, the names of her parents (if not her mother's maiden name) and the approximate location, this should be enough however some online application forms request the mother's maiden name even though this is often why you want the birth certificate in the first place! Sometimes in these instances it can be enough to write 'unknown' as the form just needs something in that text box. That is why it would be best to contact the Clare office directly through the information Brian has provided.
Once you receive the birth certificate, it will contain your grandmother's maiden name (as well as the location your mother was born). You will then be able to search for your grandparents' marriage record on www.irishgenealogy.ie which holds historic certificates (births over 100 years, marriages over 75 years and deaths over 50 years). The marriage record usually contains the names of the fathers of the bride and groom which will help you find your grandparents on the 1901 and 1911 census (www.census.nationalarchives.ie) as well as their own birth certificates through www.irishgenealogy.ie.
To research further than this, we would recommend you visit www.rootsireland.ie which contains Church baptismal records as well as contact the local rootsireland office in County Clare: Clare Heritage and Genealogical Centre,Church Street, Corofin, County Clare, IrelandTel: +353 (0) 65 6837955 Email: firstname.lastname@example.org
All the best with your search.
Ulster Historical Foundation (rootsireland centre for Counties Antrim & Down)