I too have just started researching my family history as a Gascoyne, spelt the same way as you.You never know, we may be related! My father was Frederick William Gascoyne born 18th December 1015 in Forest Gate, London.
His Father was Alfred Gascoyne born 6th November 1890 and married my Grandmother Ada Beckwith on 1st March 1913 at the register office in West Ham Essex
His father was Henry Gascoyne but that's where it all goes cold at present.
All the information is authenticated by Birth Marriage or Death Certificates, or other varifiable documentation of which I hold copies or originals
I have no idea if this is of any help, but you may pass it on to anyone who is interested along with my e-mail address.
Incidentally, I don't know whether or not you are aware of it, but Liverpool and Merseyside in general has a number of Gascoyne's using our spelling.There's a butchers shop up there called GASCOYNE's, and at one time there was a political party in Liverpool called The Gascoyne Party (they were a pro-slave party)
Good luck and keep in touch. We must be a special breed of Gascoyne.