I believe the Sig academy is in NH, that's the only "big name" school I know of in your area off the top of my head. Just look for reviews of potential trainers, talk to people who took their classes. Youtube has mountains of proof that there are lots and lots of horrible "trainers" out there.
I was an instructor for Sigarms Academy for nearly a decade (It's renamed to Sig Sauer Academy now, but it's in the same location). It's a great school, with excellent instructors and a long legacy, it was around before the explosion of defense related schools, back when there was only a dozen or so quality places in the USA. Now there are hundreds, and many great instructors, just do your research on both the type of training you want, and where who you're considering getting it from. There are forums and YT videos beyond counting you can research before you spend your $.
A shotgun is generally more effective at stopping bad guys IF YOU HIT THEM, but it can also be harder to wield if you have tight spaces or hallways/doors in your home, and generally has far less magazine capacity. A handgun is small, concealable, easy to wield and can have 18 or more rounds in the magazine, but are far weaker then a shotgun or rifle. To summarize, *Handgun = more chances to hit bad guy, but each hit is weaker* *Shotgun = fewer chances and possibly harder to hit bad guy, but each hit is much stronger* Also don't believe the Hollywood BS that you don't need to aim a shotgun, the shot spread over a typical room distance in a home defense scenario would be smaller then your fist. There is not do it all perfect firearm, everything is a trade-off.
Believe it or not AR-15 platform rifles make excellent home defense weapons also, they do more far more damage then a handgun, have double or triple the magazine capacity, very little recoil ( easy to shoot) excellent ergonomics and with a cheap red dot sight or laser it's easier to hit a target. And with the right ammo, over-penetration is not really an issue either. Although I guess an AR-15 might not be an option in Connecticut, do they have as silly of gun laws as NJ, NY and MA?
Disagree with some of this, agree with others (Ar15 part) - just because most handguns have a larger magazine capacity in no way makes them easier to use than a shotgun, in terms of getting hits. Any shoulder supported weapon is vastly superior to any handgun in terms of pointability, sight radius/length, accuracy potential, controlability, and many other factors, especially under stress conditions when the blood leaves your extremities and gross motor control takes over from fine motor control. Anytime you can pick a shoulder supported weapon over a handgun if you know you're going into a fight, you should. That's not to say that a handgun doesn't have some advantages over a shotgun or carbine/rifle, or that a handgun alone is a poor choice, however within the confines of most enclosed space fighting, including the typical home, a 20" or less barrel shotgun is easily maneuverable enough in this environment that using it's length as a reason for it being inferior to a handgun indoors is something you'll find 99.9% of the instructors out there rejecting.
Agree re: AR15s and any 556 caliber rifle, which is a good choice too, even inside the home. Due to the velocity and composition of many of the rounds, they are actually less likely to overpenetrate and strike things you don't want to hit beyond your home than many handgun or shotgun rounds, as the bullet will frequently fragment when it hits things like wood frames, furniture, many layers of walls/insulation/etc. There is some variance to this, and there are many videos online where people have tested this in old abandoned homes.
This all said, getting a handgun and some training is a great place to start. Learning to accurately shoot and manage a handgun is more difficult than a shoulder supported weapon like a shotgun or rifle as I said, so it's actually a great skill to learn first, as picking up shotgun or rifle shooting later, the same skills will transfer over, and it'll all seem much easier, again due to the nature of shoulder supported firearms.
Just pick a good quality handgun, Sig, Glock, whatever, find something that you're comfortable with, that fits your hand, and has a system that you like, be it double action/single action, striker fired, etc. Then take some training from a reputable school or instructor. You can train a lot on your own after this, others mentioned dry fire training, and this is critical, I had the 75/25 rule trained into me, that you should press that trigger training 3x more at least doing dry fire drills than your actual live fire training. There are many techniques you can use to learn proper fire/trigger control while dry firing, and train yourself to keep that sight picture perfectly alinged and stationary while moving your trigger, resetting it, and then moving it again.
Once you have some good initial training, there are many ways you can "self train" be it DVDs, books, etc However before that, in terms of learning the critical skills such as proper safety, muzzle direction control, safety ergonomics in terms of where your trigger finger goes and when, safe drawing and reholstering, and basic shooting fundamentals such as grip, stance, sight picture, fire control, follow through, reloads, clearance drills, and basic movement on a flat range, etc - you really do need professional in person instruction IMO, at LEAST a full day of it, of not more.
Also, as the OP said he's only interested in having a handgun for home defense, I'd recommend taking some specific training for that - IE don't waste your $ on a concealed carry course if you're not going to be doing that, take a basic combat/pistol course, one that focuses on indoor and close quarter shooting. There is also a ton of good info out there on preparing for a fight in your home, such as specific ways to set up your furniture to create advantages, techniques for setting up places threats will most likely go for cover, places that you can know beforehand and have spots/marks on walls for targets in order to shoot targets trying to use concealment and cover against you, and so on. I'd also highly, highly recommend you get some training on using a flashlight, as being indoors it's a coin flip if you're going to be fighting in the dark or not, your home or not, and locating, and most importantly identifying a threat in your home before engaging it is critical. Many courses will include some basic yet effective training on how to employ a white light in terms of defensive handgun shooting.
Last thing, if you're buying a handgun for home defense only, I'd recommend picking up a few other items to store with it. Body armor is very cheap, you can pick up vests from all over the place for a couple hundred bucks or even less from Surplus places - having armor on significantly increases your chances of surviving a fight, even in your own home. Store a good flashlight designed for fighting, your armor, extra magazines, a backup cell phone - even an old one that's been registered can always still call 911, and having comms can be critical if your main phone isn't with you, you can barricaded in a room without a phone, and umpteen other unforseable reasons. Also, first aid stuff - SO many people, even experience shooters, focus so much on the weapons and shooting end of defending themselves and their home, yet have NO first aid equipment or training to deal with hits friendlies may take. When a shooting happens, frequently the police won't even let emergency service/medics in until they are convinced it's secured, meanwhile you or a family member could be bleeding out waiting for help to come, from a wound you could have easily treated with the right equipment and training. Get yourself some of that training, and spend the 50 bucks it costs for a couple tourniquets, bandages, a chest decomp needle, etc. If you're getting a firearm to defend your home, spend a couple hundred extra and get the stuff you should have to increase your odds - greatly - of surviving that defense should it ever happen.
