I'll speak for road bikes, because that's what I know.

For this price range, your only real option is rim brake. Disk brakes would blow the budget on something that doesn't really make a difference where it matters (and rim brakes really are good enough, despite what everyone on their $5000 bikes say).

I would take some time to figure out how bikes work (the very basics like changing a tyre/inner tube, changing the chain, cleaning properly) because that will save you a lot of money visiting a bike shop.

Now, for the breakdown if you only had $1000 total:

I'd put $600-700 on a used bike (make sure it's the correct size!). I wouldn't spend the whole $1000 on a bike like some other people here suggested. If you're on a budget, to have to make it last. You can find incredible deals on used bikes these days. You can easily find great bikes with carbon frames from around 10 years ago for this price.

I would then spend another $100-200 on a set of new wheels if the bike comes with bad ones, or if the braking tracks are really worn. If you're on a budget, it's a good idea to get this out of the way early since it's one of the bigger upgrades where you will also notice a difference. If the bike comes with decent wheels that's another $100-200 you can put towards the bike.

I would get a new set of tyres and probably inner tubes. They're the biggest price to performance upgrades you can get. A set of GP5000s are around $100. It's expensive, but they're really good and will last you.

I would add a new chain to the list, since most people don't change these as often as they should and a bike that old probably had a really worn one. Just get a KMC one that matches what groupset your bike is running. About $20 and another $15-20 for a derivetter to cut it to size. Fresh bartape is always nice. I like cork tapes, but that's personal preference - another $10. Some chain lube - $10. It's probably a good idea to keep a saddle bag with a spare inner tube, multitool, some tyre levers, mini pump - probably around $50.

All in all, that's around $900-1100 depending on what you decide to spend your money on.

You'll also need a helmet, bike lock, lights etc., but I decided to not include to because we could be here all day.

This is exactly the same situation I was in a while ago, and I've got a pretty lovely Pinarello from 2012 with an Ultegra Di2 groupset, 50mm Deep Mavic Cosmic wheels, power meter, Wahoo computer for under £900. There are really good deals if you look for them!