僕は出来るだけ多くその地方の文化を吸収しようとつとめた。

。

I tried to absorb as much of the local culture as possible.

Literal

I [topic-は] as-much-as-possible that region [possessive-の] culture [object-を] absorb-[volitional-よう]-[quotative-と] strove.

できるだけ ('as much as possible') modifies 多く. 吸収する ('to absorb') is used metaphorically for taking in culture. ~しようとつとめる uses the volitional + と + 努める — a literary way to express earnest effort toward a goal.