What would be the first things you do to find out about an MYSQL environment you are taking over?
The pt-mysql-summary tool from Percona toolkit gives you a very nice overview of your running server.

It will tell you stuff like what the major config params are set to, what technologies ara available in your server, take a 10 second status sample, analyze your schema etc.

Also worth to run pt-duplicate-key-checker at least once, to detect redundant keys.

I'm authoring the common_schema tool for MySQL, which also offers some diagnostics: the redundant_keys view is parallel to pt-duplicate-key-checker; the data_size_per_schema view will tell you, well, the datasize per schema; the auto_increment_columns view will tell you about "free space" in your AUTO_INCREMENT columns; and more.

Watching the processes as they go, using innotop or the old-timer mytop is also worthwhile: get to know the processes that run most, the longest etc.