* Larger pot and propane burner for full-boil extract.
* Temperature control for fermentation.
* Mash tun for all grain brewing.
For me, getting precise control of fermentation temps did more to improve my beer than anything else I've done.
Getting my mash tun (cooler,) and going all-grain gave me more personal satisfaction than any other upgrade. You can make or modify any recipe so much easier; it really feels like MY beer. BTW, I use BIAB, in a cooler, not a bazooka or false bottom.
Kegs are totally unnecessary, and can really suck the money from your wallet, but they're such a luxury and convenience. I've never seen a poster say, "I tried kegging, but I'm sticking to bottles." I know I'm not going back to bottling.
I've never pulled the trigger on the big brew pot, I prefer to make 3-gallon all grain batches on my kitchen stove. But many here prefer 5- and 10- gallon batches, that's totally up to you. I may still do this but I don't really have a safe spot to work with a propane burner on my property.
There is no required order for these, change what you want to, when you want to, or don't if you're happy with what you're doing.
Water chemistry adjustments are also a big deal for many, but pointless for extract brewing. If/when you get comfortable with all-grain maybe look there.
In my opinion Partial Mash has the pains of AG and Extract brewing, without really adding much. I've done a couple PM brews, but basically stick with all grain for my 3-gallon brews, or I'll get an extract kit if I want 5-gallons. If I had a bigger pot, and a place to brew with it I'd All grain 5 gallons too.