No database created on one database program will just 'work' on the other. Asking whether two SQL database systems are compatible is kind of a mistaken question to ask. How would one expect two SQL's to be compatible?
1. By being able to transfer data from one to another?
2. By each being able to do the same or similar operations on the data?
3. By being able to simply copy the raw data files on your disk over to the other system and run them exactly the same?
For the first two questions, you're (mostly) in luck. For the third question, no two database systems ever do that.
To get data from one database to another, you just have to export and import it, and there are several ways to do that. Access is more different from MSSQL than MSSQL is from MySQL. Mainly transfers from one SQL database to another can be accomplished by outputting text data in a CSV (comma-separated) format. This is only for transferring data, not table struncture, queries, etc... To transfer structure, you need to dump the SQL commands used to create tables, edit them as needed to function on the other system, and then run them on the other system. There is a lot to learn in this area, and you have to know the differences in features supported by the databases, but a lot of it can be word-for-word copying.
This page,
has a script that create an Access module that helps export a database from Access to MySQL.
on the mysql main site, has a list of other contributed software that halps connect MySQL to other databases. One interesting method is to use ODBC, so Access can be sort of a 'front end' to MySQL.
Please feel free to ask any further questions. I have set up and configured Linux/FreeBSD and MySQL so many times I could do it in my sleep.