まだ梅雨にも入っとらんというのに…
I guess it just means "We're not even in the rainy season yet". However I can't figure out how "入る" becomes "入っとらん".
[入]{はい}っとらん is a literary or dialectal way of saying [入]{はい}っていない. Here in your example it means "hasn't entered (into the rainy season)" → "(The rainy season) hasn't started (yet)".
[入]{はい}っとらん is a colloquial contracted pronunciation of [入]{はい}っておらん.
~ておる (= て-form + subsidiary verb おる(居る)) contracts to ~とる in colloquial speech. For more on this kind of contraction, please see: ~ておく or ~とく for preparation (conjugation and nuance).
[入]{はい}っておらん is the negative form of 入っておる (= [入]{はい}っている in modern Japanese). The ん, or ぬ in its original form, is the literary negative auxiliary. For the usage and meaning of ~ておらん, please see: How to use ておらん in lines like 食べておらん?.