Guido_statche, this appears to be a software issue. The display issues are with sites like Facebook and certain news sites that use a lot of scripting. Some things work in Firefox but not Safari, and vice versa, so I don't think it's hardware related.
dca, thank you for the upgrade information! I still have Office 2008 on my own Macbook Pro running Yosemite, and although it struggles, it does actually still work (I did update to 2016 though because I need it for my side hustle) so that might not be an issue.
The computer is just being used as a "hobby machine" for my husband to poke around the internet, and I'm about to lose my job, so we don't want to put any money into it, but depending on our finances in the future, an SSD and upgraded RAM will be on the radar.