The 27-year-old former Crewe Alexandra player becomes Darrell Clarke's second new signing of the summer.
He follows this week's arrival of central defender Peter Hartley from League Two side Plymouth Argyle.
But Moore becomes the fifth Vale player to leave Burslem, having turned down the offer of a reduced contract at the up-for-sale Potteries club.
Defender Richard Duffy has moved on to non-league Eastleigh, while midfielder Michael O'Connor has already indicated that he will end his two-year association with the club and is considering offers from three clubs in England and one from abroad.
Strikers Louis Dodds and AJ Leitch-Smith have joined Shrewsbury Town, who have also signed released Vale defender Ryan McGivern.
Stoke-born Moore, who made the short move to Burslem from Gresty Road in 2014, found the net six times in 59 appearances in his two years with Vale, the majority of which came last season.
He scored 33 times in 297 games in his seven seasons with Crewe, where he graduated through the club's Academy.
Port Vale owner Norman Smurthwaite is due to name his new manager, expected to be the club's first foreign boss, early next week.
Smurthwaite had already set a deadline of next Tuesday (21 June) to announce who would succeed Rob Page.
Much-travelled Portuguese coach Bruno Ribeiro, the former Leeds United and Sheffield United midfielder, is now the bookmakers' hot favourite, ahead of Shefki Kuqi.
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Bristol Rovers have signed winger Byron Moore from fellow League One side Port Vale on a two-year contract.