I cant help but notice that the new stormtroopers in The Force Awakens are remarkably better shots than they were in Episodes IV, V, and VI. What happened?
The First Order was formed from the leftover bits of the Empire. Wouldn't these stormtroopers in the First Order still be just as bad? Was a new training regime implemented, did the First Order change up stuff for them? Or is it something else?
Although it's a common joke among avid movie fans, there's very little in-universe evidence to support the notion that Imperial Stormtroopers are as poorly trained or inaccurate as you suggest.
This question addresses it well: Are all Stormtroopers as poor soldiers as the ones in the movies?
Add in other facts such as Obi-Wan Kenobi directly referring to the solid reputation of Stormtroopers, saying:
And these blast points, too accurate for Sandpeople. Only Imperial stormtroopers are so precise.
And upon closer inspection, the Millennium Falcon was likely allowed to leave the Death Star so they could be followed to the hidden Rebel base, rather than the assumed ineptitude of the Stormtroopers failing to prevent an escape and you quickly realize that there isn't much evidence to support your claim.
As for how they compare to First Order troopers, I don't know of any source that directly addresses any drastic steps forward or backward in terms of accuracy or training.
Episode VII does more to get the viewer to respect their troopers, though, creating the perception that they're more effective. The viewer is allowed to see some of the carnage up close and personal, for example, when they slaughter the village on Jakku. And they even go as far as to introduce us to Lor San Tekka so we actually like him a little by the time he's brutally murdered.
But the Imperial Stormtroopers accomplish a similar feat in Episode IV by wiping out an entire Jawa community in search of the escaped droids. As viewers, though, we're not allowed to watch the carnage and therefore aren't necessarily given the opportunity to respect their ruthlessness in the same way.
The helmets seem to be the reason - limited field of vision and higher weight
Imperial Stormtroopers:
Its reinforced combat helmet featured an integrated comlink, audio pick-up, two artificial air-supply hoses, and a broadband communications antenna powered by a single power cell. The helmet featured built-in filtration systems that extracted breathable atmosphere from polluted environments. The helmet's visual processor assisted the wearer in seeing in darkness, glare, and smoke, though it limited the wearer's field of vision. When firing a blaster, the helmet's visor polarized against the glare. A built-in heads-up display also provided targeting diagnostics, power levels and environmental readings at the corner of the wearer's eyesight, and one could access data on various military subjects and civilian organizations on the helmets display. In addition, motion sensors alerted the wearer to any enemy the soldier might have missed. Seeking to discourage nonessential chatter which was strictly off-limits while on-duty, stormtrooper helmets recorded everything that was said by the user, sending it to monitors to review after downloading the data off of the armor's memory.
Source: http://starwars.wikia.com/wiki/Stormtrooper_armor
First Order Stormtroopers:
The stormtrooper helmets had a glossy betaplast finish, which required constant cleaning. They were also equipped with a smoke filtration system and an external tank hook-up, although the helmets could not filter toxins. Aside from providing standard protection for the wearer's head, the stormtrooper helmet had both communication and targeting systems for the foot soldiers, although quadnoculars were required for any enhanced imaging. The helmets contained polarized lenses, but lacked advanced imaging gear in order to keep the helmet's weight down. The armor was immune to fire.
Source: http://starwars.wikia.com/wiki/First_Order_stormtrooper_armor
Another IMPORTANT fact is that the FO troopers were literally kidnapped and trained from birth and the Imperials just graduated from the academy and didn't have as much extensive training
