Most important part of archery: Lots of practice time shooting.
No matter what bells and whistles you have, if you don't practice with your bow, you'll never shoot it well. Practice, practice, practice some more.
If you have the commitment to practice as you should, then the next most important thing becomes a properly set up, fit, and tuned bow. Practice enough and you can shoot an ill-fitting bow very well, so I keep this as 2nd, but having a bow that fits you and your shooting style and is set up properly for draw length, timing, and tiller, will make everything you're practicing just that much easier.
Be sure to practice a lot. Shooting fewer pins on more nights a week is more important than shooting all day one day a month.
Next thing I focus on: GOOD REST, PROPERLY TUNED! QAD HD/HDX or Code Red Ripcord. Minimal contact timing. Having a good rest, properly timed, will help you eliminate negative effects on the arrow during the shot. I don't like talking to people that drink the 'full contact' rest koolaid (whisker biscuit), but I'll leave it alone here.
Then practice more.
Good quality arrows are worth the price. I only shoot Gold Tips these days. Velocity XT's or Pro's (hunting vs 3D). Having arrows tuned for your bow, and properly set up is important. You need enough tail drag and enough FOC to produce stable flight, but also need to properly tune your arrows (nock tuning for spine) and broadheads to ensure they leave the bow well, and want to fly well for you.
Oh, and then practice.
A good sight is important, but there are LOTS of good sights on the market. More important than brand/model is the pairing of the peep sight diameter with the outer ring of the sight housing. Too small or too big and you won't be able to align your peep and front sight properly.
Don't forget to practice...
A proper stabilizer (and proper tiller balance on your bow's limbs) should probably go before the sight, but it's pretty easy to find the right stabilization, so I put it after. If you're shooting properly, the bow should want to fall forward, STRAIGHT FORWARD from your hand, following the shot.
You should be practicing instead of reading this.
All of the above assumes that you have a proper sling on your bow and proper release. Don't be foolish enough to shoot modern compound bows without them.
Did I mention that you should practice a lot?