At this budget, you won't even fit an SSD, but you can definitely squeeze in a solid 1080p gaming setup
If you're buying new, something like this would be your best bet from Google; I think they throw in free assembly
You might find a different power supply at another shop for this price, but this Intel Pentium G4560 and GTX 1050 Ti combo is a total steal for what you're spending

Sure, you could save some cash by picking up newer parts, but I wouldn't touch an old motherboard or a Hard Drive—everything else here is a great value.
I’d personally look for a used graphics card with a warranty, maybe a GTX 960, R9 380, or something similar if you can find a deal
P.S.
The CPU choice is based on the fact that it's objectively the best value up to $333. The GPU choice might shift depending on whether your budget goes up or down, but you get what you pay for. There's a decent card for every price point.
You could probably swap out the case for something a bit cheaper on this build.