Okay, here's two:
The Monitors
Tasked with exploring Time, Hypertime, the Multiverse, and eventually the broader seas of the Metaverse, and protecting it from all sorts of existential threats. Core group would consist of:
Booster Gold (father of Rip Hunter, and the grounded everyman of the group, he pilots the Waverider)
Harbinger (gatherer of the group, possible love interest for Booster, even though she may be his great-great-great-great-great grandmother)
Chunk (scientist, team brick - his body is portal to the multiverse)
The Earth-Two Speedy (he's been wandering around the Prime Earth since the Crisis. He spent 50 years as a centaur on Circe's island... that changes a guy).
Wild Card (the new name for Earth-3's Duela Dent after she gets her hands on an H-Dial)
They would have members joining from time to time, including:
Krypto (his super-canine nose can smell Hypertime!)
Teen Degaton (can he escape his fate?)
Gary Concord, the Ultra-Man (he insists that our present is "all wrong," and wants to change the past)
Moon Girl (in final battle with Triumph)
Access
Fantomah (a public domain heroine that will join their ranks for a time when they explore the greater metaverses)
The other group is a little more down-to-earth:
The Seven Wonders
A San Francisco-based group paid by a mysterious benefactor to gather once a week and compare notes. Eventually they bond over a world-threatening crisis bubbling up from under the city. They consist of:
The Human Cannonball - Ryan Chase, late of WGBS in Metropolis. Dedicated dad who uses an experimental STAR labs rig to do traffic reports and rescue innocents.
Microwavabelle - Head of the Oakland branch of Hero Hotline, she has only recently returned to active duty after accidentally burning an innocent bystander.
The Odd Man - The grandson of the original, he brings the fight to corrupt businessmen and governmental officials.
Fury I - Homeless, mentally troubled, 90 years old, and fighting a never ending battle to control the immortal spirit of vengeance within her.
Timothy Ravenwind - Last of the Ravenwind witches. Tossed out by his lodge and the military. Struggling with an alcohol problem. Looking for redemption.
Deep Blue - Personal guard of local Atlantean delegation. Master of obscure form of Atlantean martial arts. Carries a shrunken arsenal on her bracelet. Strangely perky.
Zeep the Living Sponge - At the age of 16, and alien meteorite turned Esteban Estevez into a walking sponge. Now he's nigh-immortal, can absorb anything, and is having the time of his life.
There are others.