Boxing can be a dangerous sport so it's extremely important that your trainer and gym is teaching you proper technique. Here are some tips to finding a good boxing gym.
As a general guideline, your boxing gym should be an encouraging atmosphere for learning the sport of boxing. A real legitimate boxing gym should be focused on proper technique and constant improvement. Your trainer should be able to tailor workouts to your physical abilities and explain to you the details of certain boxing techniques.
To start out, you should probably ask around through friends and search on Google Maps for "boxing" near your address. Check out several gyms and talk to a few trainers. Be honest and upfront with them as to what goals you plan to achieve in boxing (fitness, amateur competition, boxing professionally, supplementing other martial arts skills) and let them know what your budget is. Most individuals in boxing do it out of passion since the sport itself is very difficult and many people quit for that reason.

Different types of fitness centers that teach boxing:
YOUTH CENTER
These are the typical after-school programs designed to keep youth off the streets and out of gangs. There are usually all sorts of people in here - kids, women, old men, and anyone else from the neighborhood looking for a good workout. You'll see a few professionals and many young amateur boxers here. The atmosphere is real friendly and you'll get a lot of personal attention here.
The cost is minimal and most of the trainers are old veterans that love teaching people for the love of boxing. It's common for you to come numerous times and the subject of money doesn't even come up. Just show up at the gym and ask a random person how much it costs to train there. Tell them, "Hi, I'm Billy! What's your daily fee?" They'll laugh because they'll be unable to come up with a number and probably say, "Who cares? Put on some gloves... I'll be with you in 5 minutes." These gyms are good for learning but not good for developing advanced level skills.
PROFESSIONAL BOXING GYM
You'll recognize one by its smell - like a old sock unwashed for weeks. Tons of professionals and serious fighters all crowded into one open area like some kind of crazy "boxing zoo". You will get TOP-NOTCH battle-tested 110% authentic instruction. You'll see world-class sparring everyday as you warm-up. The atmosphere can be quite intimidating (imagine small guys cracking the heavy bag for hours without getting tired) but at the same time very inspirational and inspiring. The cost can range from $15 to $50 a session depending on the trainer's ego.
There's usually dozens of trainers working out of a single busy pro gym and you can shop around each day until you find one you really like. I've been able to work with former world champions (Vince Phillips) for as little as $15 per workout! The atmosphere here might seem intimidating at first because you see almost no women or children but don't worry. The fighters here are all hardcore and train to fight. Your workout WILL definitely push you beyond your limit. The trainers are old-fashion in that you'll never hear compliments on anything you do right. The sparring here can be quite tough so don't be too proud to ask the other fighter to go easy on you.
MMA GYM
MMA gyms come in two varieties - the REAL and the FAKE. The "real" MMA gyms sell themselves as the ALL-IN-ONE martial arts fight centers where they give classes on everything and supposedly "specialize in everything". Real MMA gym will offer classes on all sorts of fighting arts such as grappling, kickboxing, and boxing. Their knowledge of boxing, although good, focuses only on the basics and aspects of boxing that is useful for combat in MMA. They don't teach you the true intricacies of boxing and many fun tactics are left out because they conflict with principles of kickboxing or grappling. If you run into a FAKE commercial MMA gym, then it's really just another "fat camp" in disguise as an MMA gym.
FITNESS GYMS
These commercial gyms should definitely be avoided if you want to learn serious boxing technique. These fitness gyms are purely in it for the money and their motives are easy to spot. Their gym is very clean, often spacious, and they offer a wide assortments of price deals and training packages . The training is run more like cardio/aerobics sessions and I like to call them "fat camp". These gyms typically have the worst boxing instruction and charge you the most money. The plus side is that you'll see the most women here and so they are more comfortable for women who don't like training alongside men.

It is definitely important to see if your gym has the best interest of you. I have come accross gyms where trainning session are centered towards getting the gym "favorites" ready for there matches. This is often at expense of the beginners starting off at the gym.
Another good way to guage a gym is by talking to the fighters. You can guage the "atmosphere" gym by simply talking to the fighters. You can get to know what the great things the gym is providing and also find the negative aspects as well. Some of the fighters are waiting to transition out of the gym. Finding out the reason is the key.