No registration of handguns that you legally owned in CT but you cannot possess "assault pistols" no matter what. See the link above for the named banned assault pistols (semi automatic versions of UZIs, MACs, etc)... but full auto versions are legal... go figure. We are subjected to "child prevention access laws" here which means you get in big trouble if a child is found with one of your guns. The other bad part about ownership is one regulated firearm (handgun or so call assault weapon) every 30 days. They will prosecute you if you try. Its best to get a "collectors" designation that allows you to purchase as many as you want and is easy to get.
Transportation has been covered, but there are only a few valid reasons you can transport firearms in this state if pulled over: formal or informal target shoot, going to/from a gun show or gun smith, dog training and hunting if I recall correctly. If you become a collector (as designated by the state) you can also informally "show" your collections to others in the state.
Purchasing: No permit required, but you do have a 7 business day wait which includes a background check and paperwork recording the sale and the gun has to be on the "approved roster" if purchasing inside the state. The roster was put in place to prevent so called Saturday Night Specials in the state, which is complete BS and totally subjective. But you can bring in your .25 cal pocket pistol if you purchased it as a legal resident of another state and you don't try to sell it in this terrible state. You do need a permit to carry, but as stated above - don't waste your time because you can't use self defense as a valid reason.
Bottom line advice: Live in Virginia or Pennsylvania if you are close to the line and get your CCW (but it is not valid in MD). This state is bad with respect to firearms laws and only will get worse unless we keep up the good fight.