Ok, I don't live in either of those states, but since nobody else has given any advice yet I will try to point you in the right direction.
Check out
www.packing.org
They are more or less the comprehensive concealed carry database with lethal force laws and permitting, and purchase requirements for all fifty states. It's a really good resource. Even though it's primary focus is on CCW, it has a lot of information you will probably find usefull.
Also
www.thehighroad.org is a good website for firearms enthusiasts and they are happy to help new shooters.
I don't know the particular laws for each state, and you will have to check them yourself.
But generally speaking, you shouldn't have problems purchasing a firearm for home defense if you are not a felon, insane, too young, or in the united states illegally.
I think you will find purchasing a firearm surprisingly easy. Typically all you need is two forms of ID and payment and a background check will be conducted on the spot. In NC if you want to purchase a handgun, you have to stop by the sherif's office first and pay five dollars in lieu of a background check, but it's really quite easy. It is my understanding that NH is fairly reasonable with their firearms laws, though I have no idea about RI.
I am not going to preach about which caliber or how exactly you should store your weapons because a lot of that boils down to comfort and your unique circumstances. But I will suggest that you also consider purchasing a fast-access handgun safe (if you end up buying a handgun) because you can get to it fast enough for it to be useful. Be vary wary of trigger locks. I learned how to bypass mine when I was a very young teenager, and they can be a pain to remove, slowing you down when you need your firearm.
Also, make sure both you and your wife are comfortable actually handling and shooting whatever you settle on. Something that may feel fine to you may be too severe for her to shoot. Many establishments will allow you to rent and try out several different weapons.
Good luck