Your kit came with the most essential equipment you'll need for brewing - buckets/carboys for fermenting, a thermometer, hydrometer, sanitizer, etc. But you'll find as you journey farther down the path of the homebrewer that there are plenty of other tools and toys out there that will come in handy. I'd like to suggest some things that you may want to consider picking up to make your next batch(es) easier and more enjoyable.
Spritzer bottles (2)
These are great things to have on stand-by for at least two reasons. First, keep one filled with regular water and have it at the ready in case a boilover begins to happen in your kettle. Spritzing it quickly with cool tap water will help bust that boilover by cooling off the surface of the boiling wort. I have saved many a brewday with this tool. Your second spritzer can be filled with pre-mixed Star-San sanitizer solution for use on your equipment, hands, airlocks, etc.
Extra Muslin Bags / Bottle Caps
Buy a 10-pack of mesh bags, or spring for a reusable one. You will be kicking yourself the day you realized you've already begun the brewing process only to find you have no mesh bag for grains. Same goes for extra bottle caps.
An extra bucket will come in handy for making large batches of sanitizer solution to keep on hand or even for hauling equipment up and down stairs, in and out of the house, etc.
Extra Siphon Hose
I like to have a few pieces of hose -- cut into different lengths -- available for random tasks. For example, a four-foot section is good for emptying sanitizer or cleaner out of carboys and into other vessels or into a sink. Those carboys are heavy when they're full! Don't try to empty them by hand.
Double mesh if possible. I love my strainer for several reasons. If you are an extract w/ grains brewer or partial mash brewer the strainer is good for resting your grain bag in post-steep. On the back-end of brewing, a sanitized strainer is awesome to use while transferring wort from your kettle to its fermenter. For buckets, rest the sanitized strainer on the edge of the bucket, then pour your cooled wort through the strainer to keep a lot of that gloopy hop material out of your fermenter. This will help you salvage a greater volume of beer for bottles or the keg. For carboys, set the strainer inside the funnel.
Pegboard and Assorted Hooks
If you have the capability and the right (I'm looking at you, renters!) to install some type of metal or wood/cardboard pegboards in your homebrewery, basement, garage, utility room... do it! Pegboard is a great way to get all your nicks and nacks hanging in plain sight and ready for use at a moments notice. I have all my spoons, funnels, carboy dryers, measuring vessels and even some of my spices hanging on pegboard in my cellar room.
Irish Moss & Wyeast Yeast Nutrient
Two additives for 15 & 10 min. (respectively) left in the boil. Irish Moss helps gather proteins and helps them "bottom out," aiding in the clarification of your beer. Wyeast Yeast Nutrient has vitamins, nutrients, and other goodness to make your yeast strong during fermentation.
Yeast Starter Kit and Stir Plate
As long as we're talking about amped-up yeast, one of next-level ways to increase the health and strength of your yeast as it goes off to do battle in your carboy is to build them up in a yeast starter. A yeast starter basically multiplies the number of yeast cells by making a super-small batch of wort and giving it a gym to flex their muscles. If you plan on quickly getting into high-gravity beers (higher alcohol beers like old ales, barleywines, double IPAs) or lagers, you'll definitely want to learn to make good yeast starters sooner than later. You can make a yeast starter with just the equipment that comes in the kit, but a stir plate makes building a yeast starter even easier by using a magnetic bar in keep your starter in motion, adding oxygen to that hungry yeast.
Okay. This may not be the next thing you buy right after just getting into the hobby. But depending on how deep you fall and how quickly you become obsessed with homebrewing -- you will likely get tired of bottling all those gallons of beer. Kegs are the answer.
Pros: Wash, sanitize and fill one vessel versus 50. Carbonates quicker, so it's ready to drink sooner. Impresses the heck out of friends and family.
Cons: Not as portable as a six-pack, but you can get growlers or even a BeerGun that fills bottles as you need them for parties, picnics, etc.
These are just a few of the many items I've added to my brewing arsenal since I bought my starter kit three years ago. You'll find many more along your way. If you have some suggestions to help everyone out - leave a comment under this blog.