what are the checklist when looking at MySQL database for first time?
This question is very general and I'd suggest clarifying it with more context, but I'll give it a shot.

I would look at:


security.  What are the users that have privileges on that database?  What are the privileges (i.e. read-only on specific tables, or do they have grant option permissions)?
entities.  What are the major entities and their relationships?
performance.  What queries are resource hogs?  Are appropriate indexes in place?
denormalization.  Is the schema purposely denormalized?  If so, why and is it appropriately denormalized?
foreign keys.  Are they enforced (do the tables use InnoDB)?
size.  What are the biggest tables?  Which tables are expected to grow the fastest?