Noob database design problem
Where should I start with this?


This is a really tough question. You're basically tasked with designing, administering, and migrating to a net-new database using new technologies without much support. This isn't exactly a recipe for success at the enterprise level. In my opinion, just getting your data loaded and accessible in a mySQL database will be the first and most important hurdle. Here's a response on how to convert well formatted CSV dumps to table using HeidiSQL:
https://stackoverflow.com/questions/3635166/how-to-import-csv-file-to-mysql-table

In terms of database design, I can't really help. However getting a "copy" of what current databases look like is a great starting point. This allows you to "select into new_table query_from_source_tables". 


  How should I store the tables?


I would store them as a MySQL database called something along the lines of "source" or "legacy". You can grant other databases access as needed to these tables. 


  What ETL processes, tools should I use?


Entirely up to you. I like, generally, doing 1-off queries and then turning them into views if they need to persist/update through time. ETL is entirely up to the demand of the things which access the database. In my opinion, kick the can on this until you have things working in MySQL with your legacy dump and have a decent new design. 


  What else do I need to think about? I have been reading a lot about indexing etc, but is there anything specific to this problem that I may miss?


I wouldn't worry about indexing until you're worried about performance tuning. That usually comes after proof-of-concept. I would specifically think about sustainable database design and/or migration plans to a new database structure on the applications that leverage the old database. 

In terms of specific problems, I think you've just got your work cut out for you to redesign a database system. Focus on one "major" thing at a time, so you can focus more on what you're doing.