here are a few recommendations based on 20+ years of using a pretty basic setup. as others have mentioned, the sky's the limit on what you can spend on this hobby! in no particular order:
1). burner: I have a natural gas connection outside the house on my deck. I started with a turkey fryer burner, but graduated to a Blichmann burner which has a much higher BTU output. unless your turkey fryer burner is set up for natural gas (turkey fryer burners normally come with an orifice sized for propane), I would recommend not trying to modify the orifice for natural gas but rather buy a new burner.
2) brew kettle: even though I agree you should start with an ingredients kit, which might only require a 3-gal kettle, you will very likely graduate to all-grain brewing which will require a full-wort boil. if you plan on doing 5-gallon batches for a while, purchase at least a 10-gallon kettle (6+ gallons pre-boil volume plus headspace in the kettle for foam). I bought a heavy-duty aluminum 10-gallon kettle, plus lid, from a local restaurant supply house.
3) fermentation vessel: you can get started with a 6-gallon bucket and lid (drilled for an airlock) from your local homebrew shop. when you "graduate" to another type of fermenter, you can always use a spare bucket.
4) ingredients kit: I would strongly advise getting a kit with dry malt extract rather than liquid malt extract. you never know how fresh the LME is.
5) yeast: to get started (and maybe forever!) I would advise using dry yeast. it will work perfectly well, and nowadays (as opposed to when I was getting started brewing) there is a wide variety of dry yeasts to choose from. you will likely move on to liquid yeast, making a yeast starter, etc as you gain more experience.
6) automation: this is an expensive rabbit hole to go down. most yeasts have a fairly wide temperature tolerance. if you have a cool spot (maybe in your basement) you should be OK to get started, and not worry about a temperature-controlled fermentation chamber. I am a retired Process Automation engineer and I have resisted the temptation to use any automation for my brewing!
7). cooling: I bought a 25 foot roll of 3/8" soft copper tubing and gently rolled it around a large cylinder (somewhat smaller in diameter than my brew kettle) so the tubing wouldn't kink. since I boil and cool outside, I push a length of 3/8 ID plastic tubing over both ends and connect the tubing from the bottom of the coil to my water hose. I put the coil into the kettle for the last 10 minutes of the boil to sterilize it. again, you can upgrade this to a proper heat exchanger and pump later.
8) bottling: you will probably bottle your first few batches before jumping into kegging. I bottled for several years. to minimize the mess from spills, I opened my dishwasher door all the way (almost 90 degrees) and put the bottles on the door with the bottling bucket on the counter above the dishwasher. the spillage stayed on the opened door. then when finished, i closed the door and away went the spillage into the dishwasher drain. you might be able to snag some empty bottles for free, or for cheap, at you LHBS. sometimes home brewers that have switched to kegging donate their bottles to the store for someone to reuse. the bottles I got started with were from the days of returnable longnecks, which I just seemed to forget to return for the deposit. the returnables were very sturdy. later I starting keeping some of the one-way longnecks. if you can score some grolsch bottles from somewhere, your LHBS should have replacement caps and gaskets.
good luck!!
