Each attribute has multiple levels, the higher the level, the better. Week two, work the little beastie again. Week three rest it, and then on week four you may choose to enter the monster in a tournament battle if one is taking place at that time or you can work it again and start the whole process over the next month. You need to periodically pull up the battle schedule to see when a fight is scheduled for so your monster gets enough rest time in before you decide to fight it. You also have the option of sending the little tike to training.
Training is an intense program that focuses on improving one of the monsters attributes per session.
With the proper vitamins you can manage to squeeze a few points out on some of the other attributes at the same time to boot. On a positive note, you stand a chance of learning a new technique each time you send them to training. If it lives long enough, your monster will eventually master all of its character techniques. This will give you a very real edge in competition. Each training event consumes 4 weeks of your monsters life and they normally need to rest a week after they get back, so plan well.