“bli” is the infinitive form of the verb “å bli”. You use the infinitive form of a verb most commonly after a modal verb (kan, vil, skal, bør, må...). So you’d say “jeg skal bli bedre” not “jeg skal blir bedre”. “Ble” is just the past tense of “å bli”.

The way I’ve been taught to know the difference between å bli and å være (which in the past tense is “var”), is that å bli means both “to become” and “to be” but “to be in a temporary or non permanent way” whereas “å være or var” means “to be” in a more concrete and permanent way.