If you are intending to do things like delete and search, a database will make life a lot easier.
How many tables you use depends on how much needs to be normalized. If many 'chats' include many 'activities', you need to use 3 tables in a relational database as you cannot represent many to many relationships between two tables. This appears to require 2 tables as many 'activity' is associated with one 'chat'.
You don't really have a lot of information to work with here, so I don't think I can really suggest much else.
As of PHP 5.5, the MySQL library has been officially deprecated. It is recommended to move to either MySQLi or PDO libraries for your mysql connectivity. See here for help choosing which interface you prefer: http://php.net/manual/en/mysqlinfo.api.choosing.php