Captain: James T. Kirk
Although Sisko is my favorite captain in Star Trek, I personally don't see him as a good fit for the whole "Explore New Worlds" aspect of Star Trek. So I was torn between Kirk and Picard. Janeway was out of the running for her numerous, questionable diplomacy while in the Delta Quadrant, and Archer condemned a species to a slow death because Darwinism or something.
I decided with Kirk since he can represent the values of the Federation, while also being willing to take action against dangerous threats.
First Officer: Spock
Spock is a good match for Kirk, able to put a more logical and level-headed approach against Kirk's more brash or emotional nature. Plus, and as a running theme for a lot of my choices, has a diverse skillset that can also apply for sciences or other elements.
Chief Engineer: Miles O'Brien
A very competent engineer, who knows how to work with others. (Sorry Torres, your petty fights with that Starfleet engineer early in the series soured me). He's able to help command officers get the best of the engineers and other staff under his command (as seen in "Starship Down".
As a bonus, he's a good soldier/combat engineer as well.
Science Officer: 7 of 9
A bit arrogant, but her skills and knowledge are invaluable. Plus, having a crew member that is former Borg can help show other species that the Federation has both suffered from the Borg, but also has the technology to reverse assimilation. Lastly, Borg nano-probes are hilariously powerful/useful.
Tactical Officer: Worf
Tougher one here. Both Reed and Tuvok are both competent and professional.
Worf, however, allows for another alien voice to be represented, and shows how even those with differing cultural backgrounds can still adapt in the Federation. Plus, Worf is a competent soldier and isn't afraid to express his concerns to his superior officers.
Communications Officer: Hoshi
It's either Hoshi or Uhura, and Hoshi has a far greater skillset with languages and such. Pretty clear cut here.
Security Officer: Odo
There are strong choices here, but Worf already has Tactical. Tuvok and Reed are good choices, but Odo has some rather unique abilities that help him with his investigations. Even when he loses his powers, he is able to identify Martok as the changeling infiltration in the Klingon government. I doubt he'd have issues adapting to security on a spaceship, considering how he handled DS9.
Operations Officer: Data
We have Kim, Nog, and Data. Although Nog is a favorite character of mine, and would add another, diverse voice to the crew, I find Data has a strong variety of talents and strengths that overcome the other two. I like Kim, but he was the buttmonkey of Voyager. He'd be good to have around for him to nearly die, only to survive due to some miracle (called plot armor)
Doctor: Dr. Julian Bashir
An excellent doctor, and also an effective researcher. His genetically enhanced abilities allow for him to be useful in a variety of tasks and skills that McCoy/Crusher could not do. The EMH is also a good choice, especially for the mobile emitter...but Bashir has an additional advantage
Bashir's interactions with Section 31 makes him a valuable asset. He's able to notice details and uncover plots (Admiral Ross and his under the table deal with Section 31 over getting a plant into the Romulan leadership) but also can initiate his own schemes as needed.
Helm/Conn: Lt. Thomas Paris
Paris is competent at just about everything he is tasked to do. Good field medic/nurse. Good shuttle engineer, etc. The other options don't stand out.
Morale Officer/Counselor/Bartender: Vic Fontane
Gotta love me some 50s music.
Replaceable Ensign: Ensign Tilly
Worked in Discovery somewhere. No idea if she'd really work here, but she's an ensign, and I'm not 100% sure what the fuck she does on Discovery, outside of helping others do things as needed.