Grain or extract?
You can get everything you need from online retailers, but if you’ve got a local homebrew shop you might decide it’s worth it to support them.
You will need a large pot: 10 gallons if you plan to brew with grain, less if you’re using extract. I like the MegaPot that Northern Brewer sells, but any all-stainless pot would do. A spigot is pretty nice at this batch size, but I’ve done without.
You need a way to heat the pot. Your kitchen stove might have enough oomph to get that much water to a vigorous boil, or it might not. If it doesn’t, you’re looking at electric heaters or taking a propane burner outside.
You probably need a bigger bag for steeping grains, or a much bigger bag if you are doing all-grain BIAB. Wilser (on these forums) sells great BIAB bags.
You’ll want a long, stainless-steel spoon.
For fermenters, the premium choice is stainless steel. I prefer a spigot so you can bottle directly from the fermenter. The SS Brewtech bucket is spendy but it’s a good product. Plastic is a perfectly good option, and the Big Mouth Bubbler and Fermonster are nice products (others on this forum like the FermZilla too.) Again, go for something with a spigot. You want at least a 6-gallon fermenter, though 7 might be nice. Avoid buckets unless you really are looking to save money (in which case they’ll make perfectly good beer.) Glass, in the long term, is simply not safe and you should stay away from it.
Your kit bottle capper is decent. You’ll need to stock up on bottles and bottle caps. A bottling wand (I prefer stainless steel, and have
this one) is a necessity.
You need a supply of sanitizer. StarSan is the way to go. A powdered alkaline cleaner (PBW) is very good to have for your stainless. Pick up a 5-gallon bucket (any one will do) to hold sanitizer and clean things. A plastic spray bottle for sanitizer is also nice.
You need a decent-capacity scale (I have
this one) for grain or extract. It’s awfully nice to have a high-resolution scale (I like
this one) for hops and (if you go down this path) salts.
You need a good thermometer. (You need to periodically check its calibration, too.) Folks swear by the ThermaPen, but I use something
cheap.
You need some way to measure gravity. Most folks here will tell you to get a hydrometer for this. I recommend a refractometer instead, which is easy to use and won’t break. (Refractometer measurements go wonky when you start fermenting and there’s alcohol present, so use an online calculator to correct your measurement.) If you’re a big spender and really like gadgets, get a Tilt too. It’s totally unnecessary, but you get insight into the fermentation process that’s nice to have, especially when you’re learning.
You don’t absolutely need an immersion chiller, but at some point you’re going to want one. Copper is good. I’d recommend the biggest one from Jaded that will fit your pot, but you can save a bit of money going with another brand.
You do not need a grain mill right now, even if you’re going to be brewing all-grain. Your local store or any online retailer will crush your grain for you.
A rack to hold bottles after cleaning is nice to have.