I have free access to a family member's camera but it really isn't the best. I think it has a 1/2" sensor though the sensor is not listed anywhere on the box or in the manual which I guess is a red flag. Should I wait and save up more money to buy a better camera or could I use this one to learn on?
You can absolutely learn on it. Start to get a hood understanding of ISO, shutter speed, and aperture. Practice composing scenes and learning how to edit
If the camera is free, then use it while saving up for a better camera.
200USD is quite a low budget for a camera here in Sweden, but I don't know where you are located.
I'd tentatively recommend the used camera market to start off with, if you go that route read up on how many exposures are resonable for the camera you are looking at and then ask the seller for that info, it is not a be all end all value, but gives an idea of how hard a camera has been used, it is sort of like a car's milage.
Some models to look for on the used market is:
Panasonic Lumix GX80 or GX85 depending on your location, it is an excellent small camera with interchangable lenses and plenty of settings to play with, note however that the rear control dial often breaks, and may need servicing.
Panasonic Lumix G80 or G85 is similar to the GX moddles but has much better ergonomics and control dials.
Sony RX100 has been released in several versions, it is an excellent small camera but without interchagable lenses.
Olympus have made cameras using the same lens mount as the Panasonic cameras I mentioned above, they have released cheaper and more expensive cameras, and are common to find second hand. I don't have personal experience with them so I can't give you detailed info about them.
Canon and Nikon are large camera makers, they make great cameras, but their entry level cameras can feel a bit cheap, if you are looking at Canon, then know that they have recently replaced their lens mount, going from EF to RF, that means that there is plenty of EF gear on the second hand market, but know that the EF cameras use an analoge viewfinder and not a digital one, they take brilliant photos, but for me who is used to a digital viewfinder with all the tools they provide I find the analoge style lacking, other really like them.
Nikon also changed their F mount to the new Z mount and like with Canon there is plenty of F mount gear on the market, F mount also means an analoge viewfinder.
Not to lecture but...I had to learn on a Nikon D50 DSLR (you can literally pick up a used one for like $50 now) and I hated it at the time, but in hindsight, I had a nasty attitude about my gear and could've learned a ton a lot faster if I'd just ditched that. I would bet if I had access to one in decent condition now I could make pretty great imagery with it with my current experience/skill level, and if I hadn't been constantly pissing and moaning about the gear I could've learned a ton of theory and stuff and made much better images with it and learned a lot more with it than I did at the time. Just pick up the gear you have on hand, and treat it like you're an apprentice learning with apprentice's tools, you can get the master's tools later on.
I say this because I recently picked up a used D60 to check and yes, I make great images with it despite its age and it's tons of fun to use it. As of two days ago I was taking really nice shots I'll be keeping with it. That with an equally old inexpensive lens attached as well.
The problem I ran into is that this camera only has 2 f-stop settings: 3 and 7. Which means I can only adjust the shutter and iso. Also it can only shoot jpg files. I didn't think I was pissing and moaning. I just wasn't sure if such a limited camera would teach me the right lessons.
Oh, my bad I was mostly ranting about myself in case that's what was going on, I was just saying that when I was (because I was)...I would've been better off just using what was on hand for whatever it was capable of. My apologies for being a jerk, and sorry.
As for what you're looking at that is a bit primitive to me to be honest but you can still learn on any camera. Try and learn composition with it until you've picked up a decent one, you can learn composition on a flipphone. If you're on a budget, I will say mirrorless is probably more or less the future but DSLR tech still does the job just like film cameras still do work just fine and you can grab a used and decent DSLR for pretty cheap these days (as well as a starter lens to go with it). I'm still using DSLR's personally.