The training room is located on the second floor of the new player building.
Here you can train your fighting skills on a trainer. You can raise easier with the Kryss's on the table to the south that is set up for each fighting skill. Also inside you will find some Lockpicking training boxes that you can double click to auto set them to lock and set the skill which will help you gain faster.
Note: Trainers will not do damage to you or your weapons, armor, or shields as well. So do not fear to wear them while training on one.