What qualities does a needle that needs to be replaced show? I'm using the needle that came on my record player that I got a few months ago, so I assume the needle is of low quality.
Why assume it's of low quality? Most modern needles are made of sapphire or diamond, so they are very durable. If the sound quality is good, your needle is probably fine.
The stylus should be shaped like a cone, but smooth with a perfectly round end, and no flattened or chipped surfaces.
If it is worn or chipped, the sound will be noticeably worse, and you'll run the risk of damaging your record. With that said, it takes a lot of plays to damage a diamond. I have my mom's old turntable from the 60's and I've only ever changed the needle once in the roughly twenty years I've owned it.
The needle is probably fine as they typically last at least 1000 hours before wear becomes noticeable, but the cartridge is most likely a garbage ceramic model. Upgrading to a cheap moving magnet cartridge can greatly increase sound quality. I'm assuming your turntable doesn't have very good specs so it's not worth it to buy anything more expensive than that.