I inherited an SQL Server 2016 SP1. I am not a DBA and work at a small company where we do not have a test environment. I know that I need to install the SP2 first. Do I then install the last cumulative? Are there any tools that would help me check what issues may come up after this update.
Do I then install the last cumulative?
Yes, you should install the latest cumulative update. Some folks would advise to install the "next to latest" version. Since the latest one just came out 2 days ago at the time of this writing, it might be prudent to wait and see if any issues arise before going there.
Are there any tools that would help me check what issues may come up after this update.
There's not a tool I'm aware of. You basically need to go through the KB articles for each update if you want to see what's included. And check the KB for the CU you're installing specifically to see if there are any "known issues" with that version. Here are the KB articles for SQL Server 2016 releases: SQL Server 2016 build versions