***ZACH NIE! SPOILER ALERT***
Star Trek: Enterprise
Enterprise NX-01
Captain Jonathan Archer - Commanding Officer
"Sub-Commander" T'Pol - Executive Officer/Science Officer
Commander Charles Tucker - Chief Engineer
Lieutenant Malcolm Reed - Armory Officer
Engisn Hoshi Sato - Communications Officer
Ensign Travis Mayweather - Helmsman
Doctor Phlox - Medical Officer

Star Trek
U.S.S. Enterprise NCC-1701
Captain James T. Kirk - Commanding Officer
Commander Spock - Executive Officer/Science Officer
Dr. (Lieutenant Commander) Leonard "Bones" McCoy - Chief Medical Officer
Commander Montgomery Scott - Chief Engineering Officer
Lieutenant Hikaru Sulu - Navigator
Lieutenant Uhura - Communications Officer
Ensign Pavel Chekov - Helmsman

Star Trek: The Next Generation
U.S.S. Enterprise NCC-1701-D
Captain Jean-Luc Picard - Commanding Officer
Commander William Riker - Executive Officer
Lieutenant Commander Data - Operations Officer
Lieutenant Commander Geordi La Forge - Chief Engineering Officer
Dr. (Commander) Beverly Crusher - Chief Medical Officer
Lieutenant Commander Deanna Troi - Counselor
Lieutenant Worf - Tactical Officer/Chief of Security
Ensign Wesley Crusher - Conn Officer
Dr. (Commander) Katherine Pulaski - Chief Medical Officer
(Lieutenant Natasha Yar - Tactical Officer/Chief of Security)

Star Trek: Deep Space Nine
Starbase Deep Space 9/U.S.S. Defiant NX-74205
Captain Benjamin Sisko - Commanding Officer
Colonel Kira Nerys - Executive Officer/Bajoran Liaison
Lieutenant Commander Worf - Strategic Operations Officer
Dr. (Lieutenant) Julian Bashir - Chief Medical Officer
Chief Petty Officer Miles O'Brien - Chief of Operations
Odo - Chief of Security
Lieutenant (junior grade) Ezri Dax - Counselor
Quark - owner, Quark's Bar and Casino
Jake Sisko
(Lieutenant Commander Jadzia Dax - Science Officer)

I started typing out the cast for Star Trek: Voyager, for completeness's sake, but then thought better of it. Like Star Trek V, it is a part of Star Trek history and must not be ignored, lest we not learn from our mistakes, but for today, I'm happy and not worried about the future of my beloved franchise; so, why ruin my good humor? Thus...

Star Trek: Odyssey
U.S.S. Ulysses NCC-74297
Captain Elisabeth McKenna - Commanding Officer
Commander Benicio Torres - Executive Officer
Lieutenant Commander Sovok - Tactical Officer/Chief of Security
Liuetenant Commander K'rena Singh - Chief Engineering Officer
Lieutenant Nick Locarno - Conn Officer
Lieutenant Daniel Kim - Operations Officer
Emergency Medical Hologram "Doc" - Chief Medical Officer
Doctor Kes - Medical Officer
Lieutenant (junior grade) Neelix - Security Officer

As for what comes after Enterprise, I think the two best ideas would be either Star Trek: Federation, a show set immediately after Enterprise, exploring the 22nd century, the early days of the United Federation of Planets, and the aftermath of the Earth-Romulan War, or Star Trek: New Frontier, a show set aboard the Enterprise-E or -F in the very late 24th century, fifteen or twenty five years after the Dominion War.