On the opening page of the Club's first minute book it is recorded, under the heading " Chess Club ", that " a meeting was held at Gisson's Hotel on Saturday, October 26th, 1895.  (...) At this meeting, it was decided to ask the Literary Society, then established at Barnfield House, whether and on what terms the Society would set apart a room for the Club ".  A week later, " at Mr. Widgery's studio ", it was decided to accept a tentative offer by the Society and to go ahead. 

On 16th November, 1895, at Barnfield House, in what was already described as the Club room, a meeting of seven formed the Club.  It was to be called Exeter Chess Club; (officers) were to be elected annually in October; members were to meet " for practice and matches " on three evenings a week, and the annual subscription was to be five shillings (half a crown for members of the Literary Society).  The first President of the Club was C.J. Lambert. (...)

At the annual meeting of 1896 it was stated that the Club had 21 members, of whom 12 were members of the Literary Society.  Matches had been played and won against the Exeter Working Men's Society and the YMCA.  (...) Of the Club income of £3.15s.0d, £2.10s.6d had been paid for five sets of chessmen and three boards, " an expensive but necessary outlay ", and two more boards were required.

RJH

"This is the earliest reference to the Exeter Club I've found — Peachey's Everybody's Guide to Chess and Draughts (1896) refers to Exeter as having even at that date ' an excellent chess club

— Chris Ravilious

Match: Exeter v Plymouth.

This match, the first played between these clubs, was contested on May 18th at the Literary Society’s Rooms, Exeter. The meeting evoked considerable interest, and as the Plymouth Club has been established some years, and the Exeter Club is practically in its infancy, having seen three seasons only, the visitors were expected to prove the victors. These anticipations were not fulfilled, as after an exciting encounter Exeter won by 5 to 2

.— BCM June 1898

[...from IM Mike Conroy of Burnley, Lancs. via Phil Trussler & Bob Jones – thanks to all!]