I believe that you will need at a minimum:
MLT (I have a 10 gallon Igloo with false bottom)
7.5 gallon pot (or bigger) and burner
That's about it. You can use the 22 quart pot to heat up sparge water. If you batch sparge, no reason to get the second cooler or a sparge arm. My first AG set up (which I still use for decoction mashes and small batches when I'm not using my HERMS) is this:
I KNOW it's ugly! I did eventually get a nicer pot, and added a 5 gallon cooler for an HLT. There are some great pots out there- polarware, Blichmann, etc. But think about your goals. Are you going to want a keggle? What about doing 10 gallon batches? I wouldn't buy an expensive pot until you knew what you wanted to have in the future. An inexpensive turkery fryer can be used, and then you can use the 30 quart pot later on to heat sparge water, if you want.
You can do AG with a bare minimum of equipment. I'd start with an "upgradable" MLT. I think a 10 gallon MLT would be great- for "big" beers on 5 gallon batches, or medium OG-sized 10 gallon batches. That would be my first investment. Then the pot and burner. Make that decision based on your future goals. While an HLT and second cooler isn't a waste of money, I still rarely use an HLT if I'm batch sparging. It's just quicker to heat and dump the water into the MLT.