Advice For A New Senior DBA
Like you've already suggested, as soon as I got there I'd perform an inventory of all SQL Servers before then spending time running the latest version of sp_Blitz® – Free SQL Server Health Check Script and auditing the results. 

Depending on the results I'd then probably look to get in and standardise Ola Hallengrens SQL Server Maintenance Solution. You can then concentrate on running checkDB, index maintenance and getting backups offsite, security (logins, linked servers) etc.

Once the data is safe, I'd then look to either bring in a third party monitoring solution or build a basic one myself. At a minimum, I'd start capturing wait stats every minute or so.