The best players place their Monster Academy outside of the main infrastructure of their base. Most players will ignore them when they attack, and if destroyed, the player will not lose any resources. The only downside to this is the long repair time. Upgrading monsters should be based on both preference and effectiveness. Monsters not used often should be the last to train, for it will consume both time and putty. Players often upgrade tank monsters, such as Ichis and D.A.V.E.s, to increase health and attack, and offensive monsters, such as Banditos and Project-Xs, to increase speed and attack. Some of the more developed players have also trained their Zafreetis to heal more and have better speed.
WARNING: Once a monster's training is completed, it is irreversible. When upgrading, always check the putty cost and stat gains the monster will have, and be absolutely sure you will want it before doing it. Also check if the resulting goo costs are/will be too high.
EDIT: You use the Monster Academy to train your monsters to a higher level, which may increase goo costs, health, attack damage, and heal (Zafreetis and Vorgs only). Production time may increase or decrease, while housing space does not change except for Pokeys, which after training levels 4, 5 & 6 takes up less space. It should be placed in a well-defended area, as it takes a longer time to repair, and if it is more than 50% damaged, any training will stop.
Bug: When upgrading a monster, the goo cost can go up. If this monster is cued for production, the goo value will go up at no extra cost. Using this, you can bump up your stored goo by nearly 30% for each monster upgraded. This is mostly useful for the more novice players and those who have just recently started the game.