Walk around with monster in your batallion and it will naturally lay eggs as you walk around. After a few days, the eggs will hatch and a new monster of the same genre will be born (possibly a higher tiered type tho. i.e. Hyudra and Tiamatats from catching a Hydra)

Equip someone with "Monster Skill" squire support skill. And have that person be near a monster, and it will unlock hidden abilities of that monster

Also, Tiamats are prolly one of the best monsters you can bring into battle, the 3rd tier of Hydra