Cameron Bancroft was trapped lbw by Joe Leach with the first ball of the match.
But Chris Dent (65) led the recovery and Taylor changed the pace of the game with a stylish unbeaten 118 from 149 balls, sharing a 96-run stand with Phil Mustard (50).
Seamer Ed Barnard was the pick of the Worcestershire bowlers with 4-67.
Gloucestershire centurion Jack Taylor told BBC Radio Bristol:
"Of my five first-class hundreds that one probably gave me the most pleasure because we were facing a sticky situation when I walked out to bat. That made it all the more satisfying still being there at the close.
"We have talked about taking bowlers into their fourth and fifth spells because batting definitely becomes easier when that happens.
"All the lads enjoy Cheltenham and I love batting on the ground. I still need 30-odd to beat my best score, made here against Northants a couple of years ago, but our tail-end batsmen are capable of sticking around."
Worcestershire assistant coach Matt Mason told BBC Hereford & Worcester:
"At 85-4 after the first session we were happy. We wanted to bowl first on the wicket because it was a bit greener than pitches usually are here.
"It looked a good decision at lunch, but after that the wicket settled down and they batted pretty well.
"There was frustration in getting them seven down (for 234) and not being clinical enough to finish off the job."
Jack Taylor scored a buccaneering century as Gloucestershire reached 343-8 on the first day of their County Championship match with Worcestershire.