GIK Acoustics is a great place to learn about room treatment. They will also offer you FREE advice about what would help the most. Of course they will give you a long list of their own products, but you can ask them to prioritize AND you can crib from their advice to augment with some of your own DIY stuff if you cannot afford all their stuff.
For an AVR, on a budget, this is the best deal going:
https://www.costco.com/yamaha-tsr-700-7.1-channel-network-av-receiver.product.100686465.html
You can get the same unit under a slightly different model number from places other than Costco, MSRP at $500. And note that Costco sometimes has this below their normal $400 price during sales (recently it was $330).
This has a phono preamp, good room correction, good power and features for the price.
As I say, if you can go up to $1k then there are some Denon and Marantz options to consider. But this Yamaha is the current price/value leader in the marketplace.
If I had $1k to improve things and didn't have an AVR, I would spend half on this Yamaha and half on room treatments from GIK (or DIY if I was handy and had some time since the money will go a bit further that way).