First at all, Magento system has a built-in for cleaning up log information.

If you go to

System > Configuration > Advanced > System > Log Cleaning

You can configure your store to automatically clean up these logs.

But if you have already a very big database, the best option is cleaning the big tables. To do that try these SQL queries:

TRUNCATE dataflow_batch_export;
TRUNCATE dataflow_batch_import;
TRUNCATE log_customer;
TRUNCATE log_quote;
TRUNCATE log_summary;
TRUNCATE log_summary_type;
TRUNCATE log_url;
TRUNCATE log_url_info;
TRUNCATE log_visitor;
TRUNCATE log_visitor_info;
TRUNCATE log_visitor_online;
TRUNCATE report_viewed_product_index;
TRUNCATE report_compared_product_index;
TRUNCATE report_event;
TRUNCATE index_event;

There is another table that can be excessively large, it is core_url_rewrite and it seems that it's not a good idea to delete its values for SEO reasons. You can find more information about it here

In this tutorial you can find a really easy script to clean Magento database and cache files Magneto Log & Cache Maintenance Script