I smelt burning in my room one day coming from my computer, shortly afterwards it shut off. Now I've realized that something was clearly damaged because when I try to turn on my computer on it will light up but instantly shut off every time. Could this be a problem with the PSU or is it something bigger?
Probably - sniff and see if it the smell is stronger around the PSU. Do you have another power supply you can plug in and test? You can take another machine and just plug in the motherboard and graphics to perform a test, if you confirm the PSU.
If it is not obviously the power, look for scorch marks on all your components, especially around the connections. If you find something suspicious, and it is not the mobo or cpu, then unplug the offending component and see if you can start the machine and at least get it to post.