Please detail: are you
1) Considering using MySQL versus Access as the database for storage?
2) Or considering Access versus SQLyog as a 'frontend' for MySQL?
In case of 1) we really cannot advice you here. That would depend on what kind of application (a web application or not for instance) and what programming language you want to use, whether it is a multiuser application or not, what platform you wnat to use (obviously you cannot use Access if you would ever deploy the database on a Linux server) etc.
In case of 2) I would clearly prefer a 'true' MySQL client with no dependency on ODBC and similar for most work. That does not exclude the option that from time to time you may connect to MySQL with Access for utilization of specific Access functionalities, like the Access 'report' feature (though I'd actually prefer OpenOffice/Base w. JDBC rather than Access w. ODBC in this particular scenario. The MySQL JDBC driver seems to me to work better and be better maintained than the ODBC driver).