Guys,
I found the answer. Because I'd set the 'Region' field to be 'Text' field type, Subtype Field could not list it (for some funny reason, its as though its suppose to pick only 'Interger/Numerical' fields from ones attribute table). Anyhow, then again, I had to create a domain for each of the 'Districts' in the File Geodatabase. And so there was a domain called 'Medway' with coded values; Rochester and Gillingham, and another Domain called 'Chatham' with coded values; Newington and Key Street....... etc.
In the feature class polygon, under the subtypes tab, I selected the Region field (which is not an interger field) from the drop list in Subtype Field and started listing the different Regions, that is, code= 0 and Description= Medway; code =1 and Description=Chatham etc.
Then for the Default Values and Domains section: for each Region, say Medway, I would select the corresponding 'Domain' from the Domain drop list.
And so now, when i start editing my feature class, i get to the attribute field and from the drop down i select my Region from the Region list, the next field 'District' automatically becomes populated with the Districts of the Region I had earlier selected.
I hope this helps someone out there as it did to me.
Many thanks
Kwadwo