you can definitely use phpMyAdmin to do the job - IF your databases are not too large. phpMyAdmin chokes on imports or exports if they are too large. anything under 2MB should be fine. phpMyAdmin understands a variety of formats. I would recommend exporting SQL - structure AND data to an SQL file.
mysqldump also works. try googling for this...i can't recall the exact syntax, but i google for it all the time and a page comes up which describes how to get data in and out using mysql command line stuff.
if you use phpMyAdmin then go to a particular database (or table) and click the 'EXPORT' tab. You'll see some options there that should be quite useful. if they have phpMyAdmin on the destination machine, then you should be able to recognize some complimentary functions in there.