This with an imersion chiller, and a chest freezer with a temp control for fermenting in. Seriously, thats it. I would brew my first few batches, and see what else from there.
He's got a chiller listed on there. I totally disagree with the swamp cooler being enough for accurate temp control.
If I had 800 dollars at the beginning of this journey and someone to tell me how to spend it, I would start with temperature control. You have a freezer, that's great. Get a controller (40-100, depending on what you want.) Then, I'd get a stir plate and an erlenmeyer flask so I could do a yeast starter (50 bucks if you go cheap, which is totally fine.)
I'd get a couple of carboys (40ish each.) Better Bottles and the newer 6 gal PET bottles out there are just fine, too.
I'd get two used kegs, a co2 tank, and a regulator (about 100 for the whole setup used if you can find it.) You can serve from picnic taps for starters (20 bucks.) You need a second fridge at this point, but they're easy to come by.
I'd build a picnic cooler mash tun with a copper manifold (60-80 bucks.)
I'd buy a ten gallon pot with a ball valve, or one that I could install a valve in. The prices can vary, but if you want long term and you still have money, why not get a good one? (We'll go 150 bucks!) You will need a second vessel that can hold hot water for sparging. I'd say a 10 gallon igloo cooler is fine, or a second pot will do. Igloo is a LOT cheaper. 40 bucks.
You have a burner. That's great.
Get a hydrometer or two (20 bucks), a sample tube (3 bucks) a wine thief (10 bucks), some star san for sanitizing (10 bucks), something to stir your mash (I love heavy paddles, so 20 bucks), and it never hurts to have a bucket around for sanitizing things. And a spray bottle. (20 more bucks total.)
That's about 660 dollars.
OH! You need a wort chiller. 60 bucks for that.
720 bucks.
A bottling bucket is about 20 bucks. You can get used bottles and clean them. You'll need a wing capper and some caps. Should be another 20 bucks.
760 dollars. What am I missing? Ingredients for your first batch can be cheap enough that you should be able to get all you need from what I've listed.
I've bought all of this and much more. Can you make good beer for less? Probably. Would I go back to making beer without a freezer, a good way to make starters, and kegs? No, I would not. If you're passionate about the project, and you see yourself going at it full bore, that's my list. If you have 800 bucks to blow, that's rad and I think you should. Go economy! My next big purchase will hopefully be a triple wide fridge/freezer for fermenting, so it isn't always cheap and there's always more you can do.
I just wouldn't make beer with less than I've stated above knowing the quality I can achieve now.
Of course, I'm obsessive and I will talk to you for days on end about process and everything I do if you'll let me. My picnic cooler mash tun looks like a pile of ****, and I will continue to use it until it somehow breaks, but the things I put in it and get out of it are all dependent on my knowledge of what to do and when. The most important thing to remember is to have fun. That's what it should be all about.