hello, I'm new to subeta; can anyone help me telling me how can I increase my pet's tier; and how can I beat bottwing bomber? Ugh always draw.

Welcome :) Tiers depend on a sum of your pet's stats (strength+defense+health+speed). There's a list of how much total you need for each tier at this link: http://www.battlecoliseum.net/starting-out

Buttwing Bomber is what I guess you can call a "Glass Cannon" challenger in that he hits harder than opponents of a similar difficulty but also is easy to take out with his low health and lack of defenses. You will want to keep some sort of heal up while attacking him with a nicely damaging weapon. An infinite use healer like Red Rreign Staff of Lore or some varieties of Chelon gems would help with pairing up with a bomb weapon like Krad Bomb or Happy Bomb. If you have 64 intelligence points, you could equip Champion Scroll (likely cheaper to make at the Secluded Shrine in the Sacred Lands than to buy outright) to heal 15 while being able to use two weapons.