Get rid of directed energy weapons (other than small point defense ones) and add more torpedo tubes (which can fire scientific probes of the same diameter as the weapons). 10 or so along the bow; 2 dorsal and 2 ventral pointing forward; 2 aft. That's 16. Tubes are nothing special. Make them all automatic/robotic controlled. Magazines can be protected and deep in the ship; the torpedoes we see are fairly small, so they can carry lots of them.
Add more armor plating; enough so many of the "alien of the week" direct energy weapons can't breach the hull.
Needs a bigger medical bay and more medical personnel.
Shuttles can have hardpoints to externally fit the torpedoes the ship fires/specifically designed munitions for various tasks.
Allow shuttles to dock externally so that they can carry more if need be (recessed area of the hull that can piggyback five shuttles or so).
An aft minelayer (much bigger diameter than the torpedo tubes). Large yield proximity/magnetic devices for use against cloaked Romulan vessels and highly agile Klingon Birds of Prey. Probably don't carry the mines on normal expeditions due to safety concerns.
That'll pretty much contend with most of the military threats they encounter until they get to the tech level as seen in USS Defiant from In a Mirror, Darkly.
Don't mistake these increases in combat effectiveness measures as pushing it more towards a military mindset -- they're just tools that the crew can use when they're attacked (and they always seem to be). A military mindset comes from the actions, not the items.
Try and get the Vulcans to give up their improved tech in regards to scientific exploration. Adapt that to the ship.
Keep on working the transporter and the theory behind it. The more they use it, the more experience and insight into the things that can go wrong (if you don't use it, you don't know what is unforeseen).
That's what I'd do to the NX-class to begin with.
Its speed seems to be fine, really.