MySQL and Access are both DBMSs (Database Management Systems), so they work "approximately" the same way... but there are some significant differences.
First of all, Access is not Client-Server, while MySQL is. That means that you don't have to use an ODBC Bridge to remotely connect to MySQL : the MySQL server runs and awaits client connections. This is the common way REAL DBMS work (Oracle, Sybase, SQL Server, MySQL...).
Then, you will notice that SQL is different in Access and in MySQL, especially concerning SQL functions. This is true for all DBMS : they all have different proprietary SQL syntax, "almost" ANSI 92 compliant.
Concerning Asp or Php, I think Asp can connect to a MySQL database, so you should be able to use your old pages, provided you change one or two SQL queries. However, I'm not sure, so you'd better check that. Anyway, Php is quite easy to learn if you already know Asp.