What I use most are the free tools MySql has posted;
http://dev.mysql.com/downloads/gui-tools/5.0.html
MySQL Administrator will do the backups and much more
MySQL Query Browser is great for doing work on your database
Also GeorgeS has a free backup tool;
http://66.159.225.58/eqemu/eq.html
with MySQL Administrator you specify what you want to backup, select these tables;
account
character_
character_backup
faction_values
inventory
player_corpses
player_corpses_backup
sharedbank