I pick up used Trek's, Cannondales, Fuji's or Specializeds for under $100 and fix them up for anywhere from $20 to $50, or just buy them ready to go for around $150.
Some people might complain if I buy a used bike for $75 and put $75 in it, I have $150 into a $125 bike. Boo-hoo. My $150 bike worth $125 has 2 new tires, 2 new tubes, fresh grease, a new front derailler, handlebar foam, brake pads, etc., etc., and it is exactly what I wanted.
Their theoretical $125 bike existed for $100 at one time on CL on the other coast and was bought 25 minutes later, and will never be available, to me, when I want it/need it. Not to mention, that bike needed new tires and had a frozen cable that wasn't on the listing.
Try a bunch of bikes at a bike store, then hang out on CL and Facebook Marketplace (seems to be the new hotness, at least in my area) until you see what you want. Then, jump on it, quickly.
NO2 gives good advice except for the brakes. Bikes cannot stop faster than their tires can hold traction. If you need to improve an older bike's braking, just put new, quality tires on it. Comparing the braking of a bike with old tires and old pads to a new bike is pointless. You're not going to find many cheap bikes out there with hydraulic brakes, don't waste your time.
One more thing, inspect older Aluminum bikes very carefully for cracks, or just pass on them altogether. Many cheaper Aluminum bikes are very heavy, heavier than their steel counterparts, and Aluminum frames (and especially forks) can fail suddenly and without warning.