You'll definitely want to invest in a good pair of bag gloves. These are made to protect your hands while you beat a heavy bag, so it's best not to be cheap here or you'll end up buying a new pair after only a few weeks.
Hand wraps are important if you want to protect all of the tiny bones in your hands as well as your wrists (you do). When wrapped correctly, hand wraps provide protection for your knuckles as well as preventing your wrists from spraining or breaking while you throw heavy punches. They're also great for extending the life of your boxing gloves. Wraps collect the sweat and blood that would otherwise drip into your gloves.
A heavy bag is only required if you want to practice your boxing routine at home. If you only plan on training in the gym, there's certainly no reason to invest in a heavy bag. However, there's nothing quite as cathartic as throwing around a few punches when you get home...
Quality boxing headgear is essential if you plan on doing live sparring. It will take a lot of hits, so it's essential that you choose something that is going to last. Headgear protects you from cuts and scrapes, it will not protect you from repeated punches to the head. You can still black out, get a concussion, or any other head-related injury with headgear on. It will alleviate some of the blow, but not enough to prevent injury resulting from repeat hits.
Other than boxing gloves, a mouth guard is probably the most essential piece of boxing equipment for beginners. When sparring a mouth guard will protect your teeth from being knocked out, as well as your lips from splitting open after a jab to the face.
A must if you plan on doing any sparring. No one wants to get hit where it hurts, so this is an essential piece of equipment for boxing beginners or otherwise.
We saved shoes for the end because it's really up to you whether you decide to buy boxing boots or not. They're really not required for beginners, especially if you're not doing any fighting in the ring. However, if you want to show up to the fight looking like the best of ‘em, it might be worth the investment.