It's kinda a mix and match process.
With a budget, you sort of find the best balance you can. Hopefully, this will help.
Here's a selection of manufacturer refurbished receivers, with a factory warranty, from a very reputable source with a good return policy. I've purchased several large items from them and have never had a problem. Any of them will be a step-up from that HTIB, but I have a soft spot for Denons.
As for Subs, the ones Fuzzy mention are all good but start at about $500 and that will take a major portion of your budget right there. The VTF-1 is the least expensive option. It's great and will guarantee an eviction, but with shipping, it's $500.
Velodyne makes some very good, less expensive subs, and has for many years, you might want to consider, particularly on Amazon. Figure $300 upwards.
Now, if you want to be a bottom feeder, I just picked up one of
these to replace my trusty old $600 Velodyne VA-1210 sub I bought 14 years ago. I figured it would be a temporary solution until I could wrangle up enough cash form one of them thar $500 big boys. But, in all honesty, after living with this for a few weeks, I'm not in any rush to replace it.
So, here are a few options: GOOD receivers (including Onkyos) for anywhere from $100 (or so) on up and some subs in the $200 - $500 range.
Whatever is left can go towards your speakers, maybe even a matching center channel and surrounds if you budget wisely.
Give it a little thought.