Hooker Herring deputised for Best on numerous occasions last season, while Trimble is Ulster's most capped player.
"Rob and Andrew are two outstanding individuals who have the respect of everyone within the organisation," said Ulster Director of Rugby Les Kiss.
"The shared captaincy model will give us more flexibility and continuity."
Winger Trimble has amassed a record 206 appearances for Ulster, having made his debut for the Irish province against Cardiff back in 2005, while Herring joined Ulster in 2012.
Best hinted that he may relinquish his position following the team's 30-18 Pro12 semi-final defeat by Leinster at the RDS in May.
"I'm delighted they have agreed to take on the captaincy. In the current landscape it's hard to guarantee that one player will be available all of the time," explained Kiss.
"Rory and I have been in discussions about this since the end of last season and we felt it was the perfect time for him to assist us in developing this new leadership structure. His experience of captaining both Ulster and Ireland last season will continue to be invaluable for us as a group.
"We have strong experience across our squad that will support Rob and Andrew by driving standards off the field and assisting with tactics and decision-making in training and in match situations."
Commenting on his appointment, Herring said: "To be able to share the captaincy with someone like Andrew is massively exciting, and together, with the support of the many other leaders in the squad, we can make strides to deliver the success that the players, staff and supporters strive for."
Trimble was equally proud to be named as captain: "To be named as captain of my home province is extremely humbling and this is without doubt one of the proudest moments of my career."
Ulster will open their new season with a Pro12 game against the Dragons at Kingspan Stadium on 2 September.
The province has been grouped with Clermont Auvergne, Exeter Chiefs and Bordeaux in Pool 5 of the European  Champions Cup.
Rob Herring and Andrew Trimble will share the Ulster captaincy for the upcoming season after Ireland skipper Rory Best stepped down from the role.