"Il est important qu'ils deviennent meilleurs."
Translation:It is important that they become better.
Because what is important (i.e. that they become better) is mentioned AFTER "Il est important" it is correct to use "il". On the other hand, if what is important is mentioned first, you would use "c'est", for example: "Ils devraient devenir meilleurs, c'est important." In other words, "c'est" refers to something that has already been mentioned.
I was taught that better is mieux and meilleur is best.
"Mieux" is an adverb (it is used with a verb). It is the comparative degree of the adverb "bien" (well), the superlative being "le mieux". However, in some cases it may be used as an adjective - like in better health, better condition, looking better, to express satisfaction, to say you like/dislike something/someone: "Vous serez mieux à la campagne" (You will be better in the countryside); "Ce pantalon est bien, mais celui-ci est mieux" (These trousers are good, but those are better); "Il est mieux sans moustache" (He looks better without a moustache) - it is then an invariable adjective. It is also a noun, meaning 1. improvement: "un léger mieux" (a slight improvement) 2. the best one can do: " faire de son mieux" (to do one's best).
"Meilleur" is an adjective (used with a noun, and it must agree in gender and number with the noun it describes: meilleur / meilleurs // meilleure / meilleures. It is the comparative degree of the adjective "bon" (good) → bon / bons // bonne / bonnes, the the superlative being "le meilleur" (the best) → le meilleur / la meilleure / les meilleurs / les meilleures.