Planning my brewing investments has been immensely enjoyable part of the hobby. I do have gear that is gathering dust and a bunch of gear I've given away to new brewers to make room for upgrades. You have mentioned three areas for investment and they are all valid - brewhouse, fermentation temperature control, kegging. If you stick with the hobby my guess is eventually you will invest in all of these, and if you are here a long time possibly more than once. Here are a few quick thoughts on priority and order of investment...
Kegging - priority kind of depends on how much you like/dislike bottling process and if you have a place to put a kegerator/keezer. I did about 50 batches in bottles before switching to kegs. Never really hated the bottling process itself, but managing bottle inventory (dirty, clean, aging, conditioning, ready to drink) was an issue. I like suggestion above to go with chest freezer. You can start with picnic taps. If you get tired of beer you can use if for its original purpose. If you get tired of brewing but still like beer you can get Sanke taps and use it to dispense commercial kegs.
Brewhouse - I'd hold off on major expense here until you are sure you like making beer and have a pretty good idea of batch scale you will be happy with. I went from 5 gallon to 11 gallon to 18 gallon batches over my brewing history up to now. If I'd spent a whole lot on 5 gallon system that would have been wasted. Other issue is where you will be brewing and how you will power your brewhouse. Right now you are a kitchen brewer...do you plan to move out of the kitchen? For power do you want to go electric or propane (or maybe natural gas...love my NG).
Fermentation temperature control. For me that has been really important. I built a controller around batch 4 (an old fridge controlled by an STC-1000) and seemed to make an immediate and dramatic improvement in my beer. I've prioritized temp control ever since but stuck with fermentor in fridge/freezer rather than go the glycol route. Pros and cons either way happy to debate but for a new brewer I might suggest another solution...kveik yeast. Not clear this yeast actually needs much effort in temperature control...and what it needs will tend more on keeping it a little warmer than room temperature which is cheap and easy to do.
So here is my suggestion...
set up a keezer (chest freezer) without damaging the freezer for now...picnic taps inside the freezer, budget regulator and tank, a couple used kegs
low cost brewhouse to deliver 5.5 gallon batches...probably an inexpensive 10 gallon pot with a valve (or add a valve easy enough), a turkey burner and a bag.
experiment with kveik yeast ... control temps with a fermwrap and an inkbird.
I think you could probably have all three and stick within your budget. Anyway
@IPAMike, welcome to the hobby, good luck thanks for the thread--this sort of planning has always been for me a fun part of the hobby.