This site records the family history of William Williams (1809 – 1841) and descendants, from Wales to New Zealand and for some, on to Australia.
Despite the statement in the text above, birth and marriage certificates are now available for free on-line.
Clicking on many of the links below will open the findmypast.co.uk site. Create a free account, there’s no need to pay for any plan, once you have a free login, you can view the certificates that are directly linked from this site, or search for others.
Certificate for the Wedding 2 Feb 1827 for Mary Griffiths Llanelli St Elli Carmarthenshire- (also Llanelli death place of Edward)
The Wedding record shows they both signed with a cross rather that their own signatures, which could imply they could not write.
Also, the certificate says “with consent of the parents”. There where two witnesses, Samuel Williams – presumed to be William’s Brother, and John Thomas – unknown relation – perhaps Mary’s Grandfather.
Mary’s mothers family seems to have been from Llanelly. 
A possible baptism record is on findmypast.co.uk for 12 Jun 1809 for William Williams (possibly Wm Will I) in St Peters church, Carmarthen, South West Wales.
A baptism record for Mary Griffith 29 Mar 1810 also in Carmarthen, with parents Thomas and Mary.
Regarding birth places:
One possibility (currently the most likely) for Groes Fawr is in Marshfield (Welsh Mearun or Meyryn- a possible mis-spell of Meuryn) near Newport / Cardiff, which had a Groes Fawr up until 2012. (shown on old map)
Not too far away is Tredelerch Tn-delarch / Tref Delerch (another name for Rumney)- Cardiff – Thomas Williams’s Birthplace.
William I was recorded on various baptism records for his children, as a labourer, who moved around for work (probably on farms). However in researching where the rest of the family came from, they are mainly based in the South and South-West of Wales, and all the locations around Newport / Cardiff area fit with other family members birth and death places.
Another possible (but less likely) location:
Abergwyngregyn location of the district of Meuryn -birthplace of many of the children (unless it is a mis-spell of Maerun).
Possible Groes Fawr -Aber Falls – Birth of several children.
Marshfield is Mary James recorded birth district near Newport / Cardiff, historically part of Monmouthshire.
The reference to Mary James seems to be incorrect for the wife of William Williams I. It seems that it was Mary Griffiths, who married William Williams 2 Feb 1827 (not 18 Dec 1830), and after his death married Robert Rose 1844 in NZ. They were 17 and 18 on Feb 1827. William II was born in 1829.
Groes Farm at Groes Fawr, up until 2012 (shown on old map)
Edward Williams – Pontypridd (North of Cardiff) and –Llanelli, Carmarthenshire, UK
Alternate location details:
The following links are to interesting options, that while unlikely to be the true origins of William I, make interesting reading / viewing.
Aber Falls – possible original location of Meuryn
at the falls
Meuryn Uchaf (ie Upper Meuryn)
History of Abergwyngregyn and Ty Hir – the Long House