Yes it is a good idea. Access is very questionable as a database, especially for any type of web apps. The State of Alaska Retirement section used Access years ago and had nothing but problems so I and my section are now having to rewrite everything done in Access... only in our case going over to Oracle.
To be honest, I have never tried it with MySQL but the process is more or less the same. You will need to test it out on a test server, just install it on your PC.
With our Oracle tools we usually import the access file directly in. MySQL being a freeware thing may or may not support that.
I would export the Access Database complete with the structure into 1 Access file and 1 comma delimited text file. Then try importing one of those, one or the other should work.
My opinion, you would do better with MySQL than Access. Just as an additional tip, learn the basics of SQL, it is not required but really helps.