I think that I would make the ship look something like the
Neptune Class because it makes me think of a space shuttle with nacelles. The main anti-ship weapon would be a massive foward firing laser. The other fields of fire would be covered by railguns/coilguns or some future equivalent. There would also be several types of missles available: small anti-ship missles on pop out turrets (would have to be pulled back in and manually reloaded after firing), larger anti-ship/surface missles fired from tubes much like torpedos, and a few big nukes for surface attacks or large ships launched from the spine of the ship like submarine missles. The nukes would be viewed as a last resort and provide an ethical quandry for the crew in any situation where they can be used. The ship would have very little of the technologies seen later in Star Trek.
The crew would be armed with projectile weapons based on the larger ship mounted weapons for use in ground based combat. These weapons would be "kill" only weapons. To avoid damaging the ship, shipboard combat would use futuristic tasers that would fire a miniturized barbed taser and be capable of firing multiple rounds from a clip. The taser rounds would be available in "kill" power settings and "stun" power settings. Lasers would be too big and bulky to be used as anything other than tripod based weapons.
I would make each season = ~2 years and use dialoge to establish the time between episodes.
Season 1: Establish that the ship is going on a 4 year exploration mission making a loop to end up back at earth. The ship goes out, meets new species makes some friends and enemies and occassionally trades for technology seen later in the trek universe. There should be a lot of small breakthroughs in technology during the trip. Near the end of the first season some examples of hand held laser weapons are aobtained. Season 1 ends at the halfway point of the 4 year mission.
Season 2: continues with the return mission. Same sort of stories as season 1. When enterprise reaches home, it is announced that it will undergo a major refit to incorporate the new technology discovered during the mission. Everybody is a hero and lots of promotions are expected. End with a "will the crew be broken up?" cliffhanger.
Season 3: Establish which characters remain with the ship. Establish that the human space fleet is growing rapidly as is human expansion into space. Introduce the refit enterprise, which would be a little faster and the railguns have been replaced with the new, more compact laser system, as well as improving the main laser. The hand held projectile weapons would be replaced with hand held lasers. Send the ship out on another four year mission in a different direction. Same sorts of stories as the first two seasons. Discover shield technology, but be unable to incorporate it into the ship. Near the end of the season start hearing whispers of an unnamed power nearby that might pose a threat. Many of the treaties should include mutual defense agreements. Finish season at halfway point of the mission.
Season 4: Finish return trip. Enterprise due another refit.
Season 5: Enterprise now has shields. Starts another 4 year mission, but one of the new allies from the last trip calls for help, and a human fleet is sent to their aid, beginning the earth-romulan war. The earth forces are vastly inferior in equipment, but through human stubbornness and ingenuity, and some exceptional commanders manage to slow the romulan advance. Some alien species help, but earth bears the brunt of the combat. At the end of the season, the enterprise is saved by the appearance of the Daedalus, Earth's first battleship (I would make it about twice as big as what DITL has it, and arm it to the teeth)
Season 6: Earth-Romulan War. Even with the Daedalus class, earth is still slowly losing, but forcing the romulans to take territory inch by inch. Towards the end of the season, earth starts to add more allies because of their ability to go toe to toe with the romulans, and at the end of the season, the two sides are at a stalemate.
Season 7: Earth's success has caused more planets to join their coalition and they start to push back against the romulans, there have been tremendous losses of life on both sides and by mid-season negotiations for an end to hostilities have begun to take place. The negotiations lead to the end of the war and earth turns toward getting it's coalition against the romulans to become the UFP. The series ends with the creation of the ufp, the enterprise being decommissioned, and admiral Archer on board his new Daedalus class flagship saying that some day some other ship will wear the proud name of enterprise.