Well, I'd probably have set the series at the start of the 22nd Century, not the middle. Maybe 2110 or something. Earth would've colonized the entire Solar System and now was dealing with the extrasolar colonies it sent out earlier. There would've been earlier colonies created via usage of the Sleeper Ships of the 20th Century that were independent of Earth entirely that they would deal with (perhaps their first antagonists?) and try to work out governing details with, a potential source of space pirates.
Meanwhile, the Solar System colonies would've been better treated than the extrasolar ones due to their proximity to Earth and begin demanding somewhat better governing conditions due to being spoiled by Earth's better treatment.
The Vulcans would actually be at war with the Andorians, which would explain the strained relations with Earth and their desire to keep them from further exploration since they either don't want the Andorians to gain a new ally or to further divert resources to guarding Earth from attack.
The weaponry would all be fusion powered, with a continuity nod being that future folks would call it "atomic" as an outdated misnomer. They'd use fusion-powered lasers and plutonium torpedoes. No shields, but point defense weapons to shoot down missiles and extra thick hulls to try and survive laser beams (missiles being the stronger weapon). No transporters, but shuttles.
I'll think of more later.