I've heard a lot of articles say how bad Flash is, but what can it actually do? I've played browser games that use Flash in Chrome, and now Firefox. I know I have a real version of Flash, and I haven't ever noticed anything suspicious on my computer. What can Flash do just by being run? (My Flash is up to date, but I've been using it for years, so I've used different versions at different times).
Is there anything specific I should look for? (MacBook running latest version of Mojave)
While it's a bit dated, the below linked document may help address your question. This letter titled Thoughts on Flash was written by Steve Jobs in April 2010. While it was written to address why iOS doesn't support Adobe Flash, few points are also relevant for reliability, security and performance on macOS. I have quoted them below:
Symantec recently highlighted Flash for having one of the worst security records in 2009. We also know first hand that Flash is the number one reason Macs crash. We have been working with Adobe to fix these problems, but they have persisted for several years now.
As stated by Apple, security and reliability concerns, and availability of open and standards compliant alternatives have propelled them to discourage use of Flash. Flash still continues to be available for macOS for those who need it.
