There are some evidently some very critical differences in CT and MA gun laws. I'm pretty well versed in CT gun laws having been a resident of CT for a number of years; my knowledge of MA gun laws comes primarily from this forum.
In CT you don't have to register your firearms.
We were legally required to register our AR-15's previously, but that's no longer an option.
You can't bring your AR-15's into the state; it's a felony. You can currently legally purchase "pre-ban" AR-15's in the state of CT. So, if your AR-15's are "pre-ban", (DMCDON knows a lot more about this than I do), then you may be able to do something.
And you need to check your other firearms to make sure that they're not considered "assault" weapons by the state of CT...
You also need to read up on "high capacity magazines" because you can't bring any in; the deadline for registering them has already passed. Simple possession will be a problem for you.
There are three types of "permits", an ammunition, a longarm, and a carry permit.
Ammunition lets you purchase ammunition.
Long-arm lets you purchase longarms and ammunition.
Carry permit lets you carry handgun (concealed or open - although that's another long discussion), purchase handgun, purchase longarm and purchase ammunition. My recommendation is to apply for the carry permit.
Process is to apply through your local police dept for your "temp" permit which is good for 90 days (I think), then apply for your state-issued permit. Once you have your locally issued temp permit, getting the state permit is simple. The difficulty of getting the locally issued temp permit depends somewhat upon the local authority, although there's no such thing as a LTC-A. CT is a "shall issue" state, and they pretty much have to issue you the permit, unless they can show cause why they shouldn't. Some towns will jerk you around more than others, and the process can take several months. Also, you have to take an approved pistol course, and some towns try to tack on their own "special" little requirements such as letters of recommendation. It's your call as to whether you want to jump through their hoops or not.
Here's a useful link to start reading:
http://www.ct.gov/despp/cwp/view.asp?a=4213&q=494616
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Correct on all accounts.